Tachometer reads wrong

My truck is a 2005 2500HD with Duramax & 6sp manual.
Lately, when is is cold outside, I have been experiencing a problem with the tachometer. The instrument simply reads the wrong engine
speed. There are no codes displayed on my code reader & it is obvious from the odometer & the gear I'm in that the engine is fine.
The tach seems to "change" its zero point. This morning, it was 31F outside. When I started the truck, the tach jumped to 3750 rpm. From there, the needle went up & down as expected as the engine speed changed. As the truck cabin warms up, the "zero" point will slowly change back to what it should be. The changes back to normal occur when I stop the engine & restart it.
The dealer says the problem is the tach & wants to replace the instrument cluster. The cost would be $485 parts/labor.
Is there any fix I can do myself? Could there be dirt in the gauge & it just needs to be cleaned? How do you get the dash trim off so the instrument cluster can be rempved?
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If it's a mechanical tach:
It's doing the same as the old magnet and cup speedometers did. The grease hardened, and mad the needle do scary shit.
RK

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