...and do any of the other functions of the PSE under the
seat work? Like;

1. release of rear head rest (button with just one image, not raising, then it is electrical)
2. flaps of AC/heater when you press "rest" after switching off engine.(should hear flaps moving)
3. Retractable trunk grip.
4. Lumbar support.

Is the opening/closing of door locks possible with the button on the inside of
car?

If the pump is running at all, I would have a look at the rest
of the electrical system.

br,
syljua

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1998 CL500 W140
1984 230E W123