Related to this question.
Basically this question asks if the original @Granville proposition always fails.
Is it true that for all $g,h \in \mathbb{Z}[x]$ s.t. $g,h$ are coprime and $\deg(\mathrm{rad}(gh))>2$ exists squarefree $F \in \mathbb{Z}[x,y]$ and
$$(\deg F -2) \max(\deg g,\deg h) \ge \deg(\mathrm{rad}(F(g,h))) -1$$
where $\mathrm{rad}(f)$ is the radical of $f$, the product of the irreducible factors.
We have strong experimental evidence for this.
