Perfectoid purity under flat local base change

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $(R,\mathfrak{m})\subseteq(S,\mathfrak{n})$ be a flat local homomorphism. The flat base-change conjecture. If $R$ is perfectoid pure and $S/\mathfrak{m}S$ is regular (or even Gorenstein $F$-pure), then $S$ is perfectoid pure; and if $R$ is perfectoid injective and $S/\mathfrak{m}S$ is Cohen–Macaulay and geometrically $F$-injective, then $S$ is perfectoid injective. This is expected in view of the positive-characteristic theory, but the authors do not know how to prove it.