On composition of polynomials

Question

Given two irreducible polynomials $f_{u}(x),f_{r}(x) \in \Bbb Q[x]$, can one find two polynomials or rational functions $h_{u}(x),h_{r}(x) \in \Bbb Q[x]$ or $\Bbb Q(x)$ respectively such that:$$f_{u}(h_{u}(x)) = f_{r}(h_{r}(x))?$$

Answer 1

I answered this question here. The upshot is that the answer is usually "no", and one can explicitly describe all $f_u$ and $f_r$ for which such $h_u$ and $h_r$ do exist. The proof is very difficult, and among other things relies on the classification of finite simple groups.