Primitive pairs avoiding affine hyperplanes in finite fields

Let $q$ be a prime power, let $m$ be a positive integer, and identify $\mathbb{F}_{q^m}$ with an $m$-dimensional vector space over $\mathbb{F}_q$. Let $C=\{A_1,\ldots,A_m\}$ be a set of $\mathbb{F}_q$-affine hyperplanes of $\mathbb{F}_{q^m}$ in general position, and define

$$S_c^*=\mathbb{F}_{q^m}\setminus\bigcup_{i=1}^m A_i.$$

A pair $(\alpha,f(\alpha))$ is called primitive when both elements are primitive in $\mathbb{F}_{q^m}^*$, where $f(x)=ax^2+bx+c\in\mathbb{F}_{q^m}[x]$ satisfies $a\neq0$ and $b^2\neq4ac$. Primitive-pair conjecture. If $m\geq4$, then $S_c^*$ contains a primitive $S_c^*$ pair $(\alpha,f(\alpha))$ provided one of the following holds: $q\geq73$ and $m=4$, except for $(q,m)=(83,4)$ and $(89,4)$; $q\geq97$ and $m=3$, except for $(q,m)\in\{(103,3),(107,3),(151,3),(191,3),(211,3)\}$; or $m=2$ and $q$ is large enough. The conjecture predicts the existence of primitive pairs in the complement of $m$ affine hyperplanes under the listed numerical conditions, extending the asymptotic results of Theorem 2 and explicit calculations; the exceptional cases and the phrase “$q$ is large enough” remain to be clarified or settled.