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Pull-out the DDM (really easy) and unplug C3 the Green 12-Way connector (not related to the one under the seat)
This will disconnect both the two heater switches and the indicators, start the engine and see if the seat still warms-up, ignore the fact indicators will be off.
If the seats stay cool, then it could be a faulty switch (unlikely).
More likely is the DDM itself, probably corrosion of the PCB caused by water ingress in the past, the main PCB underside is shown below.
There's a small 16-pin chip (IC6 MC54HC589) monitors the seat switches, keypad buttons, Mem1+2 and Door Key, in my jpg the two yellow lines are the switch lines going to IC6
You can see this PCB also had some corrosion, you can see the repaired silvery tracks just above both to the left and right of the yellow lines, also check around connector C5 pins on both sides.
Interestingly the components related to your issue are in and around the same areas as the PCB above.
If there is corrosion try IPA cleaner and a small short stiff brush to clean it off, any track damage will need to be repaired (a little tricky)
Not saying 100% this is your problem, but given there should be no way this can be done automatically it seems the most likely.
I tried pressing a button whilst starting the engine and the seat heaters do come on.