Examining the 2017 paper Gasoline cars produce more carbonaceous particulate matter than modern filter-equipped diesel cars published in Nature, I find no mention of diesel cars reducing particulate matter. Figure 5 shows the "estimates of the contribution of diesel and gasoline passenger cars to ambient urban PM and carbonaceous aerosol" and none of them are negative.
I don't think it's just Google Translate and my spotty German, that whole article is a bit of a broad ramble jumping from topic to topic. Synthetic diesel, American bias, nuclear power, a short screed about the term "renewable energy"... No citations.
Professor Doctor Winfried Stöcker is a doctor of medicine and a professor at a medical university. This would seem to be a case of commenting outside one's field. Take what he says with a pinch of Salz.
The most modern versions emit in regular usage less fine dust particles than they take up with the air they suck in in the average big German city.
Even if true, fine dust particles are only a single aspect of diesel pollution. One cannot then conclude that diesel engines are fine.
He gives no explanation for how this would work, but presumably the idea is the diesel car sucks in dirty air, filters and/or modifies it somehow, then expels cleaner air. We can rephrase the claim: diesel exhaust is cleaner than the air the average big German city. Is it? Let's do the math.
The European Common Air Quality Index calls 25 μg/m^3 of particulate matter "low" and 180 μg/m^3 "very high". The US considers 180 μg/m^3 "Unhealthy for Sensitive Groups". A μg is a microgram, a millionth of a gram, 0.000001 g.
Current European particulate matter limits for passenger cars (Euro 6) are quite strict at 4,500 μg/km. I'll use that as my baseline. Maybe there are cars which emit less, but I don't think it will matter.
We need to translate μg/km into μg/m^3. What volume of air does an average, modern diesel car exhaust over a kilometer? Let's use this little Fiat 500. It has a 1.3 L engine displacement, the amount of air it's sucking in and exhausting each rotation. Let's say it's driving at 100 km/h and an economical 2000 rpm.
First, let's get 100 km/hour into km/minute.
100 km/hour * 1/60 hours/minute = 1.66 km/minute
Invert that to get 0.6 minutes/km. Now we can use that to get how many liters of air a Fiat 500 uses to go a km.
2000 rotations/minute * 1.3 L/rotation * 0.6 minutes/km = 1560 L/km
1560 L/km is 1.56 m^3/km. We want μg/m^3 so invert that: 0.64 km/m^3.
Now that we have the volume of air displaced per km, we can use that to convert μg/km into μg/m^3.
4,500 μg/km * 0.64 km/m^3 = 2880 μg/m^3
A diesel car is ingesting from 25 μg/m^3 ("low") to 180 μg/m^3 ("very high"). It is allowed to put out 2880 μg/m^3, 16 to 115 times worse than the particulate density in a European city.
Please do check my math.
The author gives no citations for their claim, and no mechanism. The data at hand contradicts them. Perhaps they have overestimated the effect of diesel particulate filters. I had to guess it could only work if diesel exhaust contained less particulate matter than the air its taking in.
The density of particulate matter in diesel exhaust under very strict European standards with a very fuel efficient vehicle would be considered 16 times worse than "very high" levels of particulate matter in a "big German city" and 115 times worse than what would be considered "low".
I don't see that tinkering with the details (different fuel consumption, displacement, rpm, particulate rate) is going to gain back the necessary 100 fold change to claim that diesel exhaust will clean the air. For example, "most modern" diesel cars would have to emit just 45 μg/km.
I don't see how they can claim diesel cars will clean particulate matter from the air.