You have it half fixed already since you know what is staying on! The radio should shut off when the car goes to sleep (even if there is a problem with it) after about 15min. so that would be the next thing to check.

Now that I think about it, I think the radio is able to be activated (by pushing the power button) even with the car still asleep, but even then it should also only stay on for 15 min.

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1988 325is (purchased new) sold
2004 X5 3.0 2005 X3 2.5
2008 X5 3.0 (new to me)