Xu's asymptotic conjecture for normal Cayley digraphs

For each group $R$ of order $r$, let $S$ be an inverse-closed subset of $R$, and let $\mathop{\Gamma}(R,S)$ be the associated Cayley digraph. It is normal when

$$R\unlhd\mathop{\mathrm{Aut}}(\mathop{\Gamma}(R,S)).$$

Xu's conjecture. The minimum, over all groups $R$ of order $r$, of the proportion of inverse-closed subsets $S$ of $R$ such that $\mathop{\Gamma}(R,S)$ is a normal Cayley digraph tends to $1$ as $r\to\infty$.

This is the formal asymptotic enumeration version of Xu's conjecture and is presented by the paper as equivalent to the Babai–Godsil conjecture. The supplied material gives no resolution status.