Xu's conjecture for normal Cayley graphs

Let $R$ be a group of order $r$ such that $R\not\cong \mathrm{Q}_8\times\mathrm{C}_2^m$ for any nonnegative integer $m$. For $S\subseteq R$ with $S=S^{-1}$, let $\operatorname{Cay}(R,S)$ be the Cayley graph of $R$, and call it normal when $R$ is normal in $\operatorname{Aut}(\operatorname{Cay}(R,S))$.

Xu's conjecture. The proportion of inverse-closed subsets $S$ of $R$ such that $\operatorname{Cay}(R,S)$ is a normal Cayley graph of $R$ approaches $1$ as $r$ approaches infinity.

The source explicitly says that the digraph part had been confirmed by Morris and Spiga, whereas this graph part remained open there.