After turn off the ignition and locking the vehicle, the passenger, rear consol and trunk cigarettes refuse to turn off (assume you connected electronic devices to these sockets).

Try to: Press lock and unlock buttons on Remote Control a few times and it will turn off the circuit. I had tried a few times and it worked.
(I assume that these sockets have relay timer to switch off by itself but mine refuse to shut off once a while)

Try it and please report your finding.

 

I bet yours are turned off the next morning without doing your stuff.

Whenever we take a long trip, we run a powered coolbox for drinks (same as Mercedes offers in Europe but from a USA BMW dealer) and EVERY time we start out the power is not there until engine start with coolbox connected.

The rest of the day the coolbox stays running during stops (engine off).

Should we forget and leave it plugged in overnight it won't be running the next morning. (And not because the battery's dead!)

I think it's indeed a timer thing and I'm even more sure the car isn't going to let you drain your battery via items plugged into these sockets. It would also be pretty boring to wait to see how long the timer runs.