Five-generation conjecture for maximal subgroups of almost simple groups

Let $G$ be an almost simple group, meaning that $T\leq G\leq {\rm Aut}(T)$ for some non-abelian finite simple group $T$, and let $H$ be a maximal subgroup of $G$. Write $d(H)$ for the minimal number of generators of $H$. Five-generation conjecture. Every maximal subgroup of an almost simple group is $5$-generated.

The known general bound in the source is $d(H)\leq 6$, while examples with $d(H)=5$ are given; it is not known whether maximal subgroups of simple groups of Lie type can require more than five generators.