What model and year? As indicated in another post, in my 1992 300D, I also had what I thought was false indication, but eventually found it was the license plate lamps. I had simply overlooked them.
But a couple of years later, the routine state inspection found the license plate lamps were not working, and the lamp indicator did NOT come on. The technician tried to replace the bulb but found the bulb was OK. Instead, the wiring was at fault -- there was a break in the ground circuit to the lamp housing. I never found the ground failure, but did enough testing to conclude the circuit was open some place between the lamp and the ground point, probably in the length of cable that flexes as the trunk lid opens and closes. That looked like a bear to get at and check/replace. (It turned out the car was in an accident and totalled before I resolved the issue.)
These circumstances differ from yours but the point is that ground failures may result in false indications, or may be a source of intermittent failures.

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DavidB29
1992 300E with ASR
35 years of Diesels until now!