Fang–Li–Wang–Xu conjecture on cubic graphical regular representations

Let $G$ be a finite nonabelian simple group. A cubic graphical regular representation (GRR) of $G$ is a Cayley graph $\mathrm{Cay}(G,S)$ of valency three whose automorphism group is the right regular representation of $G$. Fang–Li–Wang–Xu's conjecture. Every finite nonabelian simple group has a cubic GRR. This conjecture concerns the existence of prescribed-valency GRRs; the paper studies it for the family $\mathrm{PSL}_2(q)$ and establishes existence there except when $q=7$, so the assertion remains open in general.