Classification conjecture for two-dimensional cliques in pseudo-Paley graphs

Let $V$ be a $2$-dimensional $F_p$-subspace of $\mathbb{F}_{p^4}$ containing $1$, and let $g$ be the primitive element used to define the graph $PP(p^4,p+1,I)$. Two-dimensional clique conjecture. The space $V$ is a clique in $PP(p^4,p+1,I)$ for some $I$ if and only if

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

where $a=g^{(p+1)k}$ and $k$ is an odd integer. This weaker conjecture is presented as computationally verifiable in polynomial time via the algorithm cited in the source, in contrast with the preceding counting conjecture; its resolution is not supplied.