Xia–Fang finiteness conjecture for cubic graphical regular representations

Let $G$ be a finite nonabelian simple group. A cubic graphical regular representation (cubic GRR) of $G$ is a Cayley graph of $G$ of valency three whose automorphism group is the right regular representation of $G$.

Xia–Fang finiteness conjecture. There are only finitely many finite nonabelian simple groups that have no cubic GRR.

This is a finiteness refinement of the refuted conjecture that every finite nonabelian simple group has a cubic GRR. The source reports verification for alternating groups, Suzuki groups and projective special linear groups of dimension two, while the general statement remains open.