When I say the STB is turned off, I mean it's still plugged in but turned off using the STB power button on the remote.
You refer to an inactivity timer setting on the STB. Do you mean the alert that pops up after 4 hours of watching the same channel and not using the remote for any reason? That's the only inactivity timer I know of, and it would not apply when the STB is turned off.
So ti sure appears that my setup is behaving differently from yours.