Maximality conjecture for the $(\mathbb{F}_q,\alpha)$ and $(\alpha\mathbb{F}_q,1)$ constructions

Let $q$ be an odd prime power, let $m\geq 2$ satisfy $m\mid (q+1)/2$, and consider the cliques in $\mathrm{GP}(q^2,m)$ produced by the $(\mathbb F_q,\alpha)$-construction and the $(\alpha\mathbb F_q,1)$-construction. Maximality conjecture. If

$$p\nmid(m-1)\qquad\text{and}\qquad 2\leq m\leq \frac{q+1}{3},$$

then both cliques are maximal. The claim generalises the known maximality result for the corresponding constructions in Paley graphs, while the extension to generalised Paley graphs is stated under these additional assumptions.