Strict clique-number conjecture for quartic Peisert graphs

Let $q$ be a power of a prime $p$ satisfying $p\equiv 3\pmod 4$ and $q>3$. Let $P_{q^4}^*$ be the Peisert graph of order $q^4$, and let $\omega(P_{q^4}^*)$ denote its clique number. Quartic Peisert clique-number conjecture. One has

$$\omega(P_{q^4}^*)<q^2.$$

The claim would rule out attainment of the trivial upper bound in these quartic cases and, according to the surrounding text, would imply the weaker conjecture that $\mathbb{F}_q$ is a maximal clique in the Peisert graph of order $q^4$. The supplied text does not state whether it has been resolved.