Character-sum bound for primitive normal values of rational functions

Let $q$ be a prime power, let $m$ be a positive integer, and let $\mathbb{F}_{q^m}$ be a finite field with characteristic $p$. Write a rational function as

$$f=\frac{f_1}{f_2},$$

where $f_1,f_2\in\mathbb{F}_{q^m}[x]$ are relatively prime, and let $n>2$ be its degree, with $n=n_1+n_2$, where $n_1$ and $n_2$ are the degrees of $f_1$ and $f_2$. Let $W$ denote the number-of-divisors function and let $\Omega$ denote the number of monic divisors of a polynomial.

Rational-function primitive normal conjecture. There exists an element $\alpha\in\mathbb{F}_{q^m}$ such that $\alpha$ and $f(\alpha)$ are simultaneously primitive normal over $\mathbb{F}_q$ if

$$q^{m/2}>(2n-2)W(q^m-1)^2\Omega(x^m-1)^2,$$

provided that $f(x)$ is not of the form $ax^ig^h(x)$, where $i$ is an integer, $1\neq h\mid q^m-1$, and $a\in\mathbb{F}_{q^m}^*$, and, if $n_1\neq n_2$, then $p\nmid n_2$.

This is presented as an extension of earlier theorems and is based on computational experiments with related rational forms. Establishing the conjectured sufficient condition requires further analysis of the behavior of primitive normal elements over finite fields.