The Beauville structure conjecture for finite simple nonabelian groups

Let $G$ be a finite simple nonabelian group. An unmixed Beauville structure on $G$ is a pair of generating systems satisfying the freeness condition for the diagonal action on the product of the corresponding triangle curves. The alternating group of degree $5$ is denoted by $\mathfrak{A}_5$.

Beauville structure conjecture. All finite simple nonabelian groups except $\mathfrak{A}_5$ admit an unmixed Beauville structure.

This conjecture was prompted by the verification of the assertion for all finite simple nonabelian groups of order at most $50000$, with the exception of $\mathfrak{A}_5$.