For those that have a sooty exhaust pipe, stock emission system and with VCDS, take the time to log your coolant temperature during your drive.
My car, for at least the last 3 years, the inside of the tail pipe has been coated with a very slight layer of soot, light enough that you can't tell by looking at it. Starting from summer of last year, the soot accumulation started to get heavier. Right about the time before I experienced engine overheating, during mid Oct to early Nov, I got my first P0401 code, cleared it and came back a week later. By that time, the soot accumulation was starting become TERRIBLE. The entire inner surface of the exhaust pipe was cover with thick layer of soot, so thick that if I wipe it with my finger, I'd have to really scrub the skin hard to clean it off. I was thinking the DFP was pretty much severely cracked, toasted and I may have to pay for a new DPF when I send the car in for the emission fix.
However, as soon as my thermostat got replaced early November, I noticed a significant reduction of soot accumulation. The reduced soot level covering the inside of the exhaust pipe was so dramatic, it was like back to when it was couple years ago. I had the 2nd thermostat installed by dealer due to some installation mistake but the soot accumulation continue to be low, and the P0401 code never once came back again between November till the day I send my car in for the emission fix early this week.
I don't know if the TDI's ECU would enrich fuel ratio during an engine overheating situation, but in racing with gasoline powertrain, that's one of the ways to reduce cylinder head temp. If this is the case, overly rich fuel mixture and improper combustion may lead to situation where the soot reduction capability of the DPF is exceeded. As soon as the thermostat got replaced, with engine temperature back to normal, the DPF albeit slightly cracked already, is able to work close to normal again.
My point is that although P0401 may very well point toward a cracked DPF, there may be other operation conditions that would increase the load of DPF thus increasing the occurrence of the P0401 warning. It's prudent to find the cause and perform the necessary repair. In my case, although the thermostat isn't easy to fix, it puts the temperature back to normal and most likely saved me from paying for a new DPF.