Rugging on the first kilometers

Hello, everybody.

I've been the proud owner of an A6 C6 (F4) BJ '06 with Tiptronic and about 172,000km on the clock for a few months.
The engine is already 349hp.

I have a strange tumbling on the first 2 - 3 km, which is bad in the beginning and then less and less until it is completely gone after about 3 km.
Of course, all this is only when he's cold and he's been standing overnight.

Rugging reminds me of a strong imbalance in a tire; is therefore speed dependent.
Even during the ride to get in the idle does not change anything.
It starts at about 50 km/h and until 100 it gets more and more intense that I don't want to go faster either.

I talked to an owner of 2 A8 4.2 who told me that he had the same phenomenon with both A8, so I see it as motor or motor. A characteristic of the engine.
He suspects the drive shafts that the fat on the shafts is still cold and tough at the beginning, and therefore the drive train has an imbalance. Doesn't seem entirely conclusive to me, though...

Does anyone have any idea what it might be about?
Would you also tap on the drive shafts? Maybe the drive shaft to the rear axle?

Thank you for your answers!
greetings