The answer is no, but I think I see where these oft-repeated 70 m deep and the related 13 minute long dive claims come from.
70 m deep claim:
The Ilardo et al. 2018 Cell genetics paper makes this claim in the introduction:
They are renowned for their extraordinary abilities, diving to depths of over 70 m with nothing more than a set of weights and a pair of wooden goggles (Schagatay, 2014)
They cite Schagatay 2014 'Human breath-hold diving ability and the underlying physiology', but that study doesn't claim the Bajau dives to 70 m. They do mention the Ama dive to 20-30 m. They do discuss the Bajau in that paper along with other traditional diving foragers, but in the same paper discuss competitive freediving practices and physiology. Table 1 includes the current records for freediving disciples at the time of publishing , including:
Discipline |
Method |
Record Male/Female |
Constant weight without fins (CNF) |
Vertical swimming without fins |
101/69 m |
So it appears the authors of the Cell paper took the female competitive vertical no-fins freediving record of 69 m to be a feat of the Bajau. This isn't the case for the breathhold though, as in the table the static apnea (that's face submerged but not moving, so will be longer than if the athlete is diving as well) records are 11:35/9:02
13 minute breathhold claim:
This one appears to come from a 2018 NPR article on the Cell study that quotes the lead author:
"I was told by one diver that in his youth he could dive for 13
minutes," Ilardo says. And the Bajau have been observed to go 229 feet
deep.
So it is an anecdote relayed to the Cell paper author during the course of the research, of what they could do back in the day, and that article repeats the 70 m (229 ft = 69.7 m) deep claim as well.