KöMaL 2023 B5340

Let $n$ be a positive integer, and let $f$, $g$ and $h$ denote polynomials with real coefficients whose degrees are at most $n$. What is the largest possible number of real numbers $x$ for which $f(x)$, $g(x)$, $h(x)$, in some order, are consecutive terms of a non-constant arithmetic sequence, assuming that there are only a finite number of such $x$?