Perfectoid purity under split finite quasi-torsors

Suppose that $R\subseteq S$ is a finite $\mu_n$-quasi-torsor: there exists an open subset $U\subseteq\operatorname{Spec}R$ whose complement has codimension at least $2$, such that $\operatorname{Spec}S\to\operatorname{Spec}R$ is a $\mu_n$-torsor over $U$. Let $(R,\mathfrak{m})$ be a Noetherian local reduced $G_1$ and $S_2$ ring of mixed characteristic. The quasi-torsor purity conjecture. If $R\to S$ is split and $R$ is perfectoid pure, then $S$ is perfectoid pure. This generalizes the preceding split quasi-étale extension result to finite quasi-torsors of general index; the analogous assertion in characteristic $p>0$ is known, while the mixed-characteristic statement remains open.