The derangement-class product conjecture for finite simple transitive groups

Let $G$ be a finite simple transitive permutation group with point stabiliser $H$, and let $C,D$ be conjugacy classes of derangements in $G$. Set

$$E(G,H)=\begin{cases}C^2\cup CD,&(G,H)=({\rm L}_2(7),S_4),\\\{1\}\cup CD,&\text{otherwise.}\end{cases}$$

Derangement-class product conjecture. There exist conjugacy classes $C$ and $D$ of derangements such that

$$G=E(G,H).$$

This is proposed as a stronger form of the derangement-width conjecture. The paper proves the asserted form for alternating, sporadic and rank-one groups of Lie type, and for primitive groups with soluble point stabilisers; the general transitive case remains open.