Now that's interesting. When you shut off the car, but stay in it, the AUX power is still on. I think it stays on for 10 minutes. When you open the door to leave, the AUX power is shut off. So it seems like the memory is dependent on the AUX power in your car, but not mine. I wonder if you might have a blown fuse for some other background power source that the HVAC memory depends on when the AUX power is shut off.

 

So I was browsing around in the Service Manual, and there is a section for "HAC Memory Function Does Not Operate". It has you check the Power Supply Circuit For Unified Meter and A/C Amp. Depending on what you find in that test procedure, it either has you replace a fuse, a harness, or the Unified Meter and A/C Amp. I'm going to assume that you want to start with the easiest check - the fuses.

You should check 10A fuses #3, #6, and #19 located in the fuse block (J/B).