Thanks for the quick reply.
No, I did the two TTs immediately one after the other. I had played another ITT just before it with Alexandre Vinokourov in Tour Mode (1h05'23" - 1st place) with the final stage of Tour Mode inbetween and also I had rode the ITT with Leipheimer before in a previous Tour Mode where I believe I set the time of 1h05'03" which is the exact same time I set on the first attempt of my example above. Could either of these two things have affected it?
I'm going to try right now with Tony Martin, a rider I definitely haven't ridden with before.
Edit: Loaded up game -> One-Off Stage -> Stage 20 (ITT) -> HTC -> Tony Martin -> 1h05'07" (1st place)
Quit to Main Menu -> One-Off Stage -> Stage 20 (ITT) -> HTC -> Tony Martin -> 1h04'16" (1st place)
Still no trophy
