No, it is not well supported, but also not totally unsupported. At least it is not as much supported as the claim suggests and not in this automated equation it suggests. But this is still quite intriguing.
There is a link between both vitamins. The pro position in favour of the claim simply states:
[Vitamin D] Signs and symptoms beyond deficiency, however, remain to be elucidated. Based on clinical and observational data, the plasma level of 25(OH)D may serve as a 'marker' to detect or define a subclinical deficiency. Levels below 50 nmol/l might be insufficient to maintain the non-bone-related activities of vitamin D. Finally, it has to be considered that all of the nonbone activities of vitamin D are in concert with vitamin A (9-cis retinoic acid). Studies combining both vitamins in sufficient amounts (cod liver oil) demonstrated a beneficial effect on the prevention of respiratory tract infections. Consequently, it should be strongly recommended to increase the intake of vitamin D and to ensure a daily intake of vitamin A as counseled.
H. K. Biesalski: "Vitamin D recommendations: beyond deficiency." Ann Nutr Metab. 2011;59(1):10-6. DOI:10.1159/000332066
But that is far from the consensus:
Although activated vitamin D and vitamin A signal through common cofactors, they compete for each other’s function. Retinoic acid antagonizes the action of vitamin D and its active metabolite. In humans, even the vitamin A in a single serving of liver impairs vitamin D’s rapid intestinal calcium response. In a dietary intake study, Oh et al found that a high retinol intake completely thwarted vitamin D’s otherwise protective effect on distal colorectal adenoma, and they found a clear relationship between vitamin D and vitamin A intakes, as the women in the highest quintile of vitamin D intake ingested around 10,000 IU/d of retinol.
Furthermore, the consumption of preformed retinol — even in amounts consumed by many Americans in both multivitamins and cod liver oil — may cause bone toxicity in individuals with inadequate vitamin D status. Women in the highest quintile of total vitamin A intake have a 1.5-times elevated risk of hip fracture. Indeed, a recent Cochrane Review found that vitamin A supplements increased the total mortality rate by 16%, perhaps through antagonism of vitamin D. Another recent Cochrane Review concluded that although vitamin A significantly reduced the incidence of acute lower respiratory tract infections in children with low intake of retinol, as occurs in the Third World, it appears to increase the risk and/or worsen the clinical course in children in developed countries.
VitaminDWiki: Vitamin D needs some vitamin A, but not too much
One of the aspects mentioned above turns the claim on its feet: with low vitamin D you have to decrease your vitamin A intake to prevent known harmful effects. If that is one of the base lines where the claim comes from? Then this is logically unsound, the reverse is not true.
More on the metabolism of the vitamins A and D are discussed here:
J. Rodrigo Mora: "Vitamin effects on the immune system: vitamins A and D take centre stage" Nat Rev Immunol. 2008 Sep; 8(9): 685–698.
Somehwat undecided. In its essence this claim seems almost to be true. Some vitamin A is needed for proper vitamin D metabolism and functioning.
But some vitamin A is needed for prevention of death after all, or less dramatic: for a functioning body, since it is quite rightly classified as a "vitamin". How much is this "some" and does a higher intake of vitamin D increase the need for retinol? That is currently unknown. What is known is that high amounts of vitamin A (retinol) are toxic and high amounts of vitamin D are toxic. Cod liver oil for example seems to contain so much of both substances that only very small amounts of the whole complex seem to have beneficial effects, if at all.
A problem is in the phrasing of the claim: "Supplemented vitamin D". Supplemented vitamin D is almost never the active vitamin but nearly always the precursor D3 (cholecalciferol), with a much higher safety level since it is converted by the body to the active form of vitamin D as needed.
John N Hathcock & Andrew Shao & Reinhold Vieth & Robert Heaney: "Risk assessment for vitamin D" The American Journal of Clinical Nutrition, Volume 85, Issue 1, 1 January 2007, Pages 6–18, https://doi.org/10.1093/ajcn/85.1.6
That supplemented vitamin A (retinol) is usually going from "hm, good idea" to "desastrous effects" really fast should be considered as well. Cod liver oil counts as a supplement here.
J N Hathcock: "Evaluation of vitamin A toxicity" The American Journal of Clinical Nutrition, Volume 52, Issue 2, 1 August 1990, Pages 183–202, https://doi.org/10.1093/ajcn/52.2.183
Reading in the claim to broadly supplement vitamins and not forgetting about cod liver oil seems not the brightest idea. Wikipedia puts it quite simple:
Though similar in fatty acid composition to other fish oils, cod liver oil has higher concentrations of vitamins A and D. According to the United States Department of Agriculture, a tablespoon (13.6 grams or 14.8 mL) of cod liver oil contains 4,080 μg of retinol (vitamin A) and 34 μg (1.360 IU) of vitamin D. The Dietary Reference Intake of vitamin A is 900 μg per day for adult men and 700 μg per day for women, while that for vitamin D is 15 μg per day. The Tolerable upper intake levels (ULs) are 3000 μg/day and 100 μg/day, respectively. People consuming cod liver oil as a source of omega-3 fatty acids should pay attention to how much vitamin A and vitamin D this adds to their diet.
A tablespoon (13.6 g) of cod liver oil contains 136% of the UL for preformed vitamin A (retinol). Vitamin A accumulates in the liver, and can reach harmful levels sufficient to cause hypervitaminosis A. Pregnant women may want to consider consulting a doctor when taking cod liver oil because of the high amount of retinol.
Fatty acid oxidation and environmental toxins content are reduced when purification processes are applied to produce refined fish oil products.
Wikipedia: Cod liver oil
To repeat: just one table spoon of cod liver oil already exceeds the tolerable upper intake limit for vitamin A! Doing that for an intermediate time span is just looking for trouble.
And the role of both vitamins in bone health alone is fantastically complicated:
Supplementation of the diet with vitamins is a common occurrence and there is debate over whether increased vitamin A intake might promote skeletal fragility. Some studies have suggested that increased vitamin A intake may decrease BMD and promote hip fracture; however, other studies have not shown increased bone loss or increased fracture risk, and in some instances, protection from bone loss by vitamin A has been suggested. Vitamin D plays a major role in calcium absorption and mineral homeostasis. Vitamin D deficiency is common and some studies have suggested that he risk of osteoporosis and fracture may increase when increased vitamin A intake occurs in individuals with low vitamin D levels. It is possible that an increased risk of osteoporosis and fracture might exist for increased vitamin A intake and/or increased intake in the face of low vitamin D, but it appears that additional in vivo animal studies and studies in humans to confirm or dispel these possibilities will be necessary before clearer estimates of risk emerge.
Increased osteoclastic resorption of periosteal bone is a well documented in vitro consequence of excess vitamin A. Calvarial bone is considered to be a good model of periosteal resorption of cortical bone and it has been established that ATRA is a good in vitro stimulator of RANKL, osteoclastogenesis, and resorption in cultured calvarial bones. In contrast, there is also ample evidence that ATRA is a potent in vitro inhibitor of osteoclastogenesis in cell culture systems employing progenitors from either mouse bone marrow or spleen, or human peripheral blood. Periosteal resorption of cortical bone has also been firmly established as an in vivo consequence of excess vitamin A in experimental animals, as well as in humans. Furthermore, available evidence suggests that excess vitamin A also decreases endosteal blood flow, reduces endosteal osteoclast formation, and promotes endosteal mineralization.
Effects of vitamin A on cancellous bone resorption are less clear. Of the three animal studies investigating cancellous bone, one has shown a decrease in trabecular osteoclasts with no change in trabecular bone mass. These data appear to be in agreement with in vitro investigations indicating inhibition of osteoclast progenitors from bone marrow; however, the two other studies have reported decreases in trabecular bone mass and, in one of these investigations, no change in trabecular osteoclasts. Thus, it appears additional in vivo animal studies will be necessary before these different observations can be clarified.
Most of the studies on effects of vitamin A on osteoblast differentiation and formation indicate that vitamin A promotes osteoblast differentiation and bone formation. Interestingly, injury induced heterotopic bone formation can be inhibited by RAR signaling by retinoids.
In postmenopausal osteoporosis, the prevailing idea has been that loss of trabecular bone was the main cause of increased fracture, but evidence is now accumulating which suggests that loss of cortical bone might be the major determinant of fracture risk in these patients. Cortical bone thinning occurring in humans as a consequence of increased vitamin A intake would also cause bone fragility, but unlike osteoporosis in the postmenopausal female, where cancellous bone loss is significant, cancellous bone loss may not play as prominent a role in vitamin A excess. In the past, DXA has been employed to measure BMD and has proven to be valuable clinically, especially in the post menopausal female, but DXA does not distinguish between cortical and trabecular bone. To what extent past use of DXA for BMD measurements using criteria normally employed for conditions like post-menopausal osteoporosis may have lead to differing outcomes in some of the studies evaluating vitamin A intake, BMD, and fracture risk is unclear, but use of more specific tools to distinguish between trabecular and cortical bone appear to be essential.
Other than for the suggestion of low levels of vitamin D, it is not known presently if there might be other specific conditions or dietary and environmental differences that might enhance bone loss in the presence of increased concentrations of vitamin A.
Valuable insight for preventing future harmful effects of vitamin A on bone might be gained by additional correlative studies of vitamin A intake. Animal studies using state of the art techniques to access site specific effects of vitamin A on bone resorption and formation, BMD, and bone fragility also seem clearly warranted, as do studies in mice with cell specific deletions of different RARs and RXRs. Genetic studies evaluating the relationships of RAR and RXR genes to bone mass and fragility fracture in the human should additionally be pursued. Moreover, prospective studies in humans using peripheral quantitative computed tomography (CT) for evaluation of specific effects in cortical and trabecular bone appear to be essential if we are to gain a better understanding of the risk that increased vitamin A intake might pose to skeletal health.
H. Herschel Conaway & Petra Henning & Ulf H. Lerner: "Vitamin A Metabolism, Action, and Role in Skeletal Homeostasis", Endocrine Reviews, Volume 34, Issue 6, 1 December 2013, Pages 766–797, https://doi.org/10.1210/er.2012-1071
One of the strongest papers on the matter – often cited by the contra position – seems to suggest that even a slightly elevated level of vitamin A (retinol) intake completely counteracts the expected effects of vitamin D when related to decreased colorectal cancer risk, whether from food or from supplements:
Mazda Jenab et al.: "Association between pre-diagnostic circulating vitamin D concentration and risk of colorectal cancer in European populations:a nested case-control study" BMJ 2010; 340 (Table on page 8) doi: https://doi.org/10.1136/bmj.b5500 (Published 22 January 2010),BMJ 2010;340:b5500
In light of the available evidence and the consensus still far on the horizon for a more detailed understanding of the relationship between intake levels of these vitamins: the results are inconclusive. So inconclusive as to judge the claim as unfounded when considering nutritional or medical science. It is an overstatement of the facts available and the claim is formulated in a way that very likely promotes dangerous assumptions and behaviour in supplement takers. Following the advice associated with the claim unsupervised and without taking into account what else with how many vitamins in it is consumed will be detrimental in most cases.