Beauville's existence conjecture for finite nonabelian simple groups

A Beauville surface is a rigid surface isogenous to a product of curves. For a finite group $G$, say that $G$ occurs if it is the group defining such a Beauville surface.

Beauville existence conjecture. Every finite nonabelian simple group occurs except $\mathfrak U_{5}$.

The text presents this as evidence toward the existence question for Beauville surfaces and notes substantial progress on the classification of which finite groups can occur. The assertion is not stated as resolved in the supplied material.