Sounds like some with the AUX VENT circuit? I remember the only time they open is in OFF & AUX VENT...but tripping when you shut down the cargo pack...maybe not. It’s likely a valve, versus a solenoid, right? What about the underfloor heat valve? Doesn’t it also close when you shut down the cargo AC?
My experience is only on Es & Hs, so not sure what differences in the J....but you didn’t say what model you’re on.