Alright....your problem got me thinking. If it only does it while you are sitting idling......does it do it in park, drive or both? Does it do it consistently or intermittently? Can you reliably make it do it? Does it happen at an exact time related to a shift event (exactly 10 minutes)? Does it happen sooner when it is hotter outside?
Could be related to computer doing some test cycle like the loose gas cap thing doesn't trigger check engine until computer does some kind of purge cycle..... If this is the case it should occur at some specific time.
If not time related, I'd look at temperature. Maybe sensor getting too hot or something like that. Can't imagine it being melted wire as once that goes bad, it would be bad all the time. These things get hot under the hood. Hood is very well insulated and doesn't get very hot. That heat has to escape elsewhere (like by the front fenders causing discoloration on some vehicles). I've noticed the heat there on mine but no color change yet....not hot where I live yet....
If it is heat related it should happen whether you are in park or drive. Clue that it is heat related might be that it happens earlier when it it hotter outside. Might look at temperatures on offroad pages and see if there is any correlation there to onset of symptom. If heat seems related, and symptom is predictable I would park it while running in park or drive (with parking brake on) and open the hood and see if that affects time to onset. Consider using a leaf blower to blow air in different areas while running the test. It might help you pinpoint the problem area.....The passenger side has a bunch of wires and stuff that looks like it could get hot. Most of the stuff I could see was well wrapped in heat insulation on mine, but given the excellent attention to detail by Jeep assemblers, some wire wrapping could've been missed on some vehicles.
If you hit pay dirt with the leaf blower, then getting an hp tuner and turning fan on at lower engine temperature might be helpful.
Or, just keep doing what you are doing and wait on dealer. Sounds like shutting it off and turning it back on stops it... Does it happen again (maybe quicker than the first time) after turning it off and back on?