I tend to avoid posting here, but this seems like a question I want to find out an answer to.

Can a natural number $n$ be nontrivially palindromic in more than $3$ consecutive integer bases?

Nontrivially means that I'm not counting one-digit palindromes.

The initial question was asked on Math.SE and holds all the progress so far and references the $3$-digit patterns, but it seems that the exceptions appearing among them prevent a full analysis of the patterns.

I would prefer for the question to be answered there with a proof that no three consecutive palindromes will ever expand to a fourth one or more. (Or a counter example)

If there is any other mathematical way to tackle a problem like this one, then I would like to be able to understand the formulation of the answer (proof) if possible, since I'm curious why this seems to be the case.