ainarssems back in post #2 hinted at a possible vacuum leak. Do you know if they investigated that possibility?
Looking at the 4.2tdi, there is a vacuum pipe (#2 & 3 in the first image) with a non-return valve from the brake reservoir to the mechanical vacuum pump. Cars that have stop/start have a pressure sensor on them (PR code: 7L8).
There is also the vacuum pump itself (second image), although I can't find a decent image of where it is located on your engine. The following video shows what it looks like, but this was on a FSI engine. It might be worth checking those areas for vacuum leaks.
https://www.youtube.com/watch?v=EvfFk3ZJP6o
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2001 A8 D2 FL 3.7 Q - LPG Conversion, Engine: AKC, Gearbox:FBB, Colour:LY5X Aqua Blue pearl effect, Interior:WJ light beige/dark beige valcona leather with burr walnut inserts
My DIY jobs -
Replaced alarm horn,
Replaced Coolant Temp sensor,
Updated RNS-D firmware,
Installed Reversing Camera and Digital TV,
Dashcam Installation,
Retrofitted Rear Blinds,
Auto-Dimming Rear View Mirror,
Chrome Boot Struts
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2008 S8 D3 5.2 V10 - Engine: BSM, Gearbox:KLW, Colour:LY7F Suzuka Grey Pearl Effect, Interior:Black/Black with Silver/Black valcona leather with carbon atlas inserts, Bang & Olufsen, Heated windscreen, Soft-close doors, TV, DAB, AMI, Quad Zone, Auto Boot, Auto dim Mirrors, Keyless entry, PDC with rearview camera, AFS II Headlights
My DIY jobs -
Fan Jet Windscreen Washers,
Retrofitted Electric Side Blinds