Maximality conjecture for the $Q_0$ and $\alpha Q_0$ constructions

Let $q$ be an odd prime power and let $m\geq 2$ divide $q+1$. The $Q_0$-construction and the $\alpha Q_0$-construction produce cliques in the generalised Paley graph $\mathrm{GP}(q^2,m)$, with their precise forms depending on whether $m$ divides $(q+1)/2$. Maximality conjecture. If

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

then the cliques from the $Q_0$-construction and the $\alpha Q_0$-construction are maximal. This extends the corresponding maximality results known for Paley graphs of square order to generalised Paley graphs under additional assumptions.