First thing to do is measure the voltage at fuses 13 and 14 (both ends) with the headlights on and combination switch on high beam.
Questions: does the high beam indicator work? Does the 'flash to pass' work with the ignition switch on?
There are NO relays associated with the headlight basic function. I assume that the mystery module you are talking about is the lamp failure module in the fuse block. You are correct, the module is looking for a voltage drop across the resistive strips in the module. I have seen a lot of problems in this type of system. As in analog circuits, you have to trace the voltage to find where it stops. This is basically an analog circuit with only 0V and 12V. I recommend every one wanting to fix this type of problem get a 12v test light. It is easy to use and read. Hook it to ground and probe a circuit that you know works. In this case, turn on the low beams and check the fuses for the low beams. If the light doesn't work, check it at the battery to make sure it works. If everything is working, now turn on the high beams and check the fuses for the high beams. Let us know what you find and we can go from there.

Paul

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84 500 SEL (307,xxx miles)