Babai–Godsil conjecture on almost all Cayley graphs being GRRs

Let $R$ be a finite group, and let $S=S^{-1}$ be an inverse-closed subset of $R$. The Cayley graph $\mathop{\Gamma}(R,S)$ has vertex set $R$ and edges determined by $tr^{-1}\in S$; a graphical regular representation (GRR) is a Cayley graph whose full automorphism group is $R$ acting regularly on its vertices. A group is generalised dicyclic if it belongs to the corresponding excluded family, and an abelian group has exponent greater than $2$ when some element has order greater than $2$.

Babai–Godsil conjecture. If $R$ is not generalised dicyclic or abelian of exponent greater than $2$, then for almost all inverse-closed subsets $S$ of $R$, $\mathop{\Gamma}(R,S)$ is a GRR.

This conjecture concerns the asymptotic prevalence of graphical regular representations among Cayley graphs. Godsil's theorem characterizes the groups admitting at least one GRR, but the supplied source does not state whether this stronger almost-all assertion has been resolved.