Almost all finite Cayley graphs are stable

Let $G$ be a finite group of order $r$. An inverse-closed subset $S$ of $G$ satisfies $S=S^{-1}$. The Cayley graph $\operatorname{Cay}(G,S)$ has vertex set $G$, with adjacency determined by $x^{-1}y\in S$; it is stable when its automorphism group has the expected form $\operatorname{Aut}(D(\operatorname{Cay}(G,S)))=(R(G)\rtimes\langle\iota\rangle)\times C_2$, where $R(G)$ is the right regular representation of $G$ and $\iota$ is inversion.

Stability conjecture. For a group $G$ of order $r$, the proportion of inverse-closed subsets $S$ of $G$ such that $\operatorname{Cay}(G,S)$ is stable approaches $1$ as $r$ tends to infinity.

This conjecture proposes that almost all finite Cayley graphs have the smallest possible automorphism groups, extending the corresponding result established in the paper for finite abelian groups and relating to the notion of most rigid representations.