Conjecture on fusion systems of Sylow subgroups of Lie type groups

Let $p$ be a prime, and let $\mathcal{S}$ be the collection of pairs $(G,S)$ such that $G$ is a simple Lie type group defined in characteristic $p$, $G$ is not isomorphic to $\operatorname{PSp}_4(p^a)$, and $S\in\operatorname{Syl}_p(G)$. Lie-type fusion-system conjecture. For all but finitely many exceptions, if $(G,S)\in\mathcal{S}$ and $\mathcal{F}$ is a saturated fusion system on $S$ with $O_p(\mathcal{F})=1$, then there is a group $H$ with $G\leqslant H\leqslant\operatorname{Aut}(G)$ such that $\mathcal{F}=\mathcal{F}_S(H)$. The conjecture predicts that, apart from finitely many exceptions and the excluded family, such fusion systems are induced by almost simple overgroups of the underlying Lie type group; the source does not resolve the remaining cases.