Improved upper-bound conjecture for clique numbers of Paley graphs

Let $p$ be a prime with $p\equiv 1\pmod{4}$, and let $q=p^{2s+1}$ for some positive integer $s$. Write $\omega(P_q)$ for the clique number of the Paley graph $P_q$. Improved Paley-graph clique-number conjecture. There is some constant $c>0$ such that

$$\omega(P_q)\leq \sqrt{\frac{q}{2}}+cp^{s-1}.$$

This would improve the available upper bounds for the clique number of Paley graphs of odd prime-power order and is presented as a possible consequence of a variant of the paper's main theorem. Its status is not resolved in the supplied text.