Canonical-subspace conjecture for maximum cliques in semi-primitive pseudo-Paley graphs

Let $X=PP(q,2d,I)$ be a semi-primitive pseudo-Paley graph with $q=p^{2rt}$, where $r$ is even. A maximum clique is a clique of size $\omega(X)$. Canonical-subspace conjecture. If

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

then every maximum clique in $X$ is an $\mathbb{F}_{p^t}$-affine subspace. This conjecture extends the known $\mathbb{F}_p$-subspace structure in the odd-$r$ case and asserts that square-root maximum cliques in the even-$r$ case have the corresponding canonical subfield structure. The source presents this as open.