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 - it references the $3$-digit patterns, but it seems that the exceptions appearing among them prevent a full analysis of the patterns and thus the question remains unsolved.

By posting here, I'm hoping to find out If there is any other mathematical way to tackle a problem like this one?