Conjecture on saturated fusion systems on groups of order p^6

Let $p\geqslant 5$, let $S$ be a $p$-group of order $p^6$, and let $\mathcal{F}$ be a saturated fusion system on $S$ such that $O_p(\mathcal{F})=1$ and $O^p(\mathcal{F})=\mathcal{F}$. The order-$p^6$ fusion-system conjecture. Either $S$ has maximal class, or $S$ is a Sylow subgroup of one of $\operatorname{PSL}_3(p)\times\operatorname{PSL}_3(p)$, $\operatorname{PSL}_3(p^2)$, $\operatorname{PSL}_4(p)$, or $\operatorname{PSU}_4(p)$. Furthermore, if $p\geqslant 11$ and $S$ has maximal class, then either $S$ is a Sylow $p$-subgroup of $\operatorname{G}_2(p)$, or $S$ has an abelian subgroup of index $p$—perhaps even $S\cong B(p,6;0,0,0,0)$—and $\mathcal{F}$ is obtained by pruning one of the fusion systems in Clelland and Parker (2010) having $\operatorname{Aut}_{\mathcal{F}}(\gamma_1(S))\cong\operatorname{PSL}_2(p)$ acting irreducibly on $\gamma_1(S)$. The second part is stated to fail for $p=5$, while computations for $p=7$ motivate the restriction $p\geqslant 11$; its general validity remains open.