Infinite-family conjecture for non-Paley pseudo-Paley graphs

For a prime $p\geq 3$, consider subsets $I\subset\{0,1,\ldots,p\}$ with $0\in I$ and $|I|=(p+1)/2$, and the associated graph $PP(p^4,p+1,I)$. Let $g$ be the primitive element used to define the cyclotomic classes. Infinite-family conjecture. For each prime $p\geq 3$, exactly

$$\frac{p^2+3}{4}$$

such subsets $I$ satisfy

$$\omega\bigl(PP(p^4,p+1,I)\bigr)=p^2.$$

Furthermore, in these graphs, every maximum clique containing $\{0,1\}$ is of the form

$$\mathbb{F}_p\oplus a\mathbb{F}_p,$$

where $a=g^{(p+1)k}$ and $k$ is an odd integer. The conjecture is motivated by the expectation of infinitely many non-Paley examples, while the source notes that direct verification is impractical and introduces a weaker conjecture for computational testing.