Conjecture on irreducible factors of rational-function iterates

Let $q$ be a prime power and let $n\geq 1$. Write

$$\bar{n}$$

for the least power of $q$ that is at least $n$, and set

$$d=\left\lceil 2\log_q n\right\rceil.$$

Let $T$ be the set of pairs $(g,h)\in\mathbb{F}_q[X]^2$ with $\deg g,\deg h\leq d$ such that the rational function $f=g/h$ satisfies the multiplicative-independence conditions from Corollary IND. Rational-function irreducible-factor conjecture. There exists $(g,h)\in T$ such that $X^{\bar{n}}h(X)-g(X)$ has an irreducible factor of degree $n$. Such a pair would yield a finite search procedure for constructing elements of high multiplicative order in $\mathbb{F}_{q^n}$. The conjecture is presented as a weaker conjecture for the general rational-function setting, and the supplied text gives no evidence that it has been proved or refuted.