Maximality conjecture for subfield cliques in generalized Paley graphs

Let $d$ be a positive integer greater than $1$. Let $q \equiv 1 \pmod {2d}$ be a power of a prime $p$, and let $r$ be the largest integer such that $d \mid \frac{q-1}{p^r-1}$. The generalized Paley graph $GP(q,d)$ has vertex set understood from the notation, and a clique is a set of pairwise adjacent vertices. Subfield maximality conjecture. The subfield $\mathbb{F}_{p^r}$ forms a maximal clique in $GP(q,d)$. This conjecture strengthens the known subfield lower bound for the clique number, which gives $\omega\big(GP(q,d)\big)\geq p^r$; the supplied text does not state whether it has been resolved.