I'm glad that our results agree, yours by the book and mine by experimentation.
It would make make sense to me if the GP stayed on after the engine was running but the MB engineers should know more about it than I do. Possibly it puts a strain on the GP that shortens their life if they are subject to the heat and pressure of a running engine when the GP are red hot.
I remember reading about another make Diesel car (I can't remember what make) that the GP came on every time the engine was at idle for lower emissions. However it seems to me that if that engine was at idle too long, the battery would go dead.
P E H