Asking for guesses?
Maybe when driving, when going over bumps, the increase in the pressure in the air shock overcomes the sealing of a small leak in an airbag. When there is no high pressure demands on the bag, the rubber of the airbag is able to keep the internal pressure ("level for months if I don't drive it").
The suspension compressor should, for a while, work to keep the pressure up. It is designed to shut off when overheating, so that would account for your suspension working fine for a while, then the compressor overheats/stops and your suspension pressure will not build back up, causing the drop. 45 minutes of the compressor running more than usual would do it; if your system holds pressure for shorter drives, it would indicate the compressor is working the whole time without shutoff and keeping the pressure up.
If the car drops more on one side than the other, the lower side most likely has the air leak. The suspension airbag is the most suspect, but it could also be in the air lines or connectors.
You will probably need to take it to the dealer, a tech with the GDS system has many testing options you do not have available. Maybe just have them do a diagnostic (without repair) so you know for sure.
2011 Granite Gray EQUUS