Yip's maximal subfield clique conjecture for generalized Paley graphs

Let $d>1$ be an integer. 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 subfield $\mathbb{F}_{p^r}$ is a clique in the generalized Paley graph $GP(q,d)=\operatorname{Cay}(\mathbb{F}_q^+;(\mathbb{F}_q^*)^d)$. Yip's maximal subfield clique conjecture. The subfield $\mathbb{F}_{p^r}$ forms a maximal clique in $GP(q,d)$. This conjecture asserts that the largest subfield producing a subfield clique cannot be extended to a larger clique. The paper resolves it in a stronger form under a sufficient-size hypothesis, while the unrestricted statement is the conjecture being addressed.