Well, a few weeks ago I broke down and bought a new battery and installed it.

The same day I also fixed a niggling issue with my rear passenger door lock.

My dogs had managed to unscrew the top plastic part and this allowed the threaded shaft to get caught under the lip of the opening in the door trim.

As a result, the door stayed locked.

Since I fixed that and installed the new battery I have had zero issues with my alarm going off for no reason.

I hope this helps somebody.

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1993 W124 300D
-297K on the clock as I type this.