Exponential-growth conjecture for irreducible factors of iterated polynomials

Throughout, let $(f,g)$ be a pair of polynomials over $\mathbb{F}_q$ that are neither critical nor $p$-critical, with $f$ irreducible and $\deg g\geq 2$. Let $N_{f,g}(n)$ be the number of irreducible factors of $f(g^{(n)}(x))$, and let $M_{f,g}(n)$ be the maximum degree of one of its irreducible factors. Exponential-growth conjecture. Either

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

or

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

The conjecture asserts that at least one of the two factorization statistics always has exponential growth; the preceding lower bound that one of them is large at each level does not by itself establish this asymptotic conclusion.