Sounds like MPEM isn't going to sleep for some reason, or something is waking it up.
Try disconnecting the DESS post and see if that makes any difference. Who knows but MPEM should go to sleep with engine off by itself even with key remaining on DESS but it's an idea of something to try.
Disconnect start/stop switch and see if this makes any difference.
If you press the start/stop switch after leaving DESS on for long time, the engine cranks and starts or does it give a long beep?
Does the engine start or crank without DESS key? (I bet no)
Does the buzzer beep two chirps when you place the lanyard key on DESS post?
Connector #1 pin 24 purple wire of MPEM should not be 12V while MPEM is sleeping.
The service manual provides a DESS/start/stop button test mode procedure, give that a try and see if it leads somewhere.
Good luck!