Hmm. That's a good question. It's almost impossible to answer with any certainty because we don't know what polymers are used, aside from the very general fact that they're all polyurethanes. (And asking for polyurethane solvents is about as specific as asking"what foods do mammals like?")
DMSO or
NMP or
DMF might work as a liquid dip. DMSO is reasonably low toxicity, the others I would avoid contact with women who are or may become pregnant. (You don't want to get any of them on your skin.) But none of them are anywhere near volatile enough for vapor polishing. My concern is that you'll soften the surface and then it will take ages to re-harden unless you have something like a vacuum chamber to extract the residual solvent.
I'm thinking heat is a better approach... heat gun or touch-up iron.