Sorry to disappoint you, but the current LP12 does not fall out of tune
I’m not disappointed, I’ve heard the later iterations ( $60k fully loaded ) and they are indeed much improved. I’ve often thought about getting another to put my Naim Aro on, for posterity. I also liked the Pink Linn with vector drive some years ago.
Try your Linn without a plinth, you can use T-slot extrusions. you might be surprised. No box is without its own inherent colourations.
As an aside I can't stand the current fad of replacing the springs with grommets - definitely out of tune literally with that "mod". I hear one note bass with grommets.