If the engine has a rought time starting WHEN WARM, but starts fine when cold, and runs fine onces started, you could be having a fuel pressure leakdown. Likely suspects are a bad injector (can go hand in hand with a rough idle), or a failing accumulator ($65 part).
This is just a hypothetical. On my w126, replacing the accumulator and the injectors at 100k miles did miracles in terms of smoothness. There was nothing wrong with the OVP.
1989 560 SEL (black/black)
2001 Audi TT Roadster (silver/grey)