The FF-subgroup conjecture for finite simple groups

Let $G$ be a finite simple group. A nontrivial subgroup $H$ of $G$ is an FF-subgroup if every generating pair of $G$ can be replaced by a generating pair containing an element of $H$.

FF-subgroup conjecture. All nontrivial subgroups of a finite simple group are FF-subgroups.

The paper has established this assertion for $PSL(2,q)$, but proposes it in general for finite simple groups; its status beyond the cases proved there remains open.