"Exactly. If I am not mistaken, after days of physical exertion, testosterone levels go up in a man."
I'm pretty sure that the organisation that tests racing cyclists will be aware of this...
Testosterone levels usually goes down after more than one hour of hard physical exertion, to rebound slowly during the next 12-24 hours.
During the tour with many days in row with all out effort, testosterone levels go down quite dramatically over the three weeks, as the recovery time is not enough to get testorenone levels back to base levels.
That is why, a testosterone shot this late in the tour would be particularly beneficial.
Cheers.