Of COURSE it popped back up - the BCM had commanded it to go up (for what every reason) and the limit switches told the BCM it was down - so the BCM put it back up.

The question here is, WHY does the BCM want it up all the time? Take the cover off the module box and inspect for chewed wiring in there (the big box next to the battery). THEN, lift up the mat in the trunk and look for water sitting under there. If both of these show no issues, I'd first guess flaky battery but you have eliminated that.

So, grap the service manual and a voltmeter and check for what voltages you have at the connector pins on the BCM that come from the dash switch - could be, as indicated before, a bad switch TELLING the BCM to raise it (but in 6 years with these cars, I've never seen that.)