Classification conjecture for triply-transitive strongly-regular graphs

Let $\Gamma$ be a graph as in Hypothesis~; in particular, $\Gamma$ is a strongly-regular graph in the setting of the paper. Complete classification conjecture. The graph $\Gamma$ is triply transitive if and only if it is one of the following: a complete multipartite graph with $n$ parts of size $m$; the cycle on $5$ vertices; the McLaughlin graph; the Higman–Sims graph; the Peisert graph $P^*(9)$, isomorphic to the Paley graph on $9$ vertices; the $n\times n$ grid for some integer $n\geq 2$; the collinearity graph of the polar space $O_6^-(q)$; or the affine polar graph $VO^\varepsilon_{2m}(2)$, for $m\geq 2$ and $\varepsilon=\pm 1$. If the two preceding triply-transitivity questions have affirmative answers, this classification follows as a corollary; without those answers, the classification remains open.