The mixability conjecture for finite groups generated by involutions

Let $H$ be a finite group generated by elements of order $2$; such a group is called $\{2\}$-generated. The involution-generation mixability conjecture. Every finite $\{2\}$-generated group is mixable. This follows from the main mixability conjecture if every such group is $2'$-simple, as observed in the paper. The source notes that finite simple groups and quotients of Coxeter groups provide examples of groups generated by involutions, but the asserted mixability statement remains open.