Finiteness conjecture for finite simple groups without cubic GRRs

Let $G$ range over finite nonabelian simple groups, and call $G$ cubic-GRR-free if it has no cubic graphical regular representation. Finiteness conjecture. There are only finitely many finite nonabelian simple groups that are cubic-GRR-free. This is posed as a problem motivated by the classification of cubic GRRs for $\mathrm{PSL}_2(q)$; the source gives no resolution of the finiteness assertion.