Two-clique conjecture for square-root maximum cliques

Let $X=PP(q,2d,I)$ be a semi-primitive pseudo-Paley graph with $q=p^{2rt}$, where $r$ is even, and assume

$$0\in I,\qquad I\ne\{0,2,\ldots,2d-2\}.$$

A maximum clique is a clique of size $\omega(X)$. Two-clique conjecture. If

$$\omega(X)=\sqrt{q},$$

then exactly two maximum cliques in $X$ contain $\mathbb{F}_{p^t}$. This conjecture strengthens and generalizes the theorem discussed immediately before it, and the source presents it as an open conjecture motivated by computational examples.