The dichotomy conjecture for the growth of irreducible-factor degrees

Throughout, let $(f,g)$ be a pair of polynomials over $\mathbb{F}_q$ that are neither critical nor $p$-critical, with $f$ irreducible and $deg g greater than or equal to 2$. Let $M_{f,g}(n)$ denote the maximum degree of an irreducible factor of $f(g^{(n)}(x))$. The growth dichotomy conjecture. One of the following holds:

$$M_{f,g}(n)\approx n;$$

or

$$\log M_{f,g}(n)\gg n.$$

This would rule out polynomial growth of intermediate degree for $M_{f,g}(n)$ and, according to the authors, a proof or disproof would help solve the preceding problem of characterizing the polynomials $g$ for which $M_{f,g}(n)\approx n$.