Conjecture on perfectoidness of Shimura varieties at infinite $p$-level

Let $(G,\mu)$ be a Shimura datum, and fix a level $K^p\subset G(\mathbb{A}^{\infty,p})$ away from $p$. The inverse limit over compact open subgroups $K_p\subset G(\mathbb{Q}_p)$ is

$$\varprojlim_{K_p}\operatorname{Sh}_{K_pK^p}(G,\mu).$$

Global perfectoidness conjecture. This inverse limit is representable by a perfectoid space. Perfectoidness at infinite $p$-level is a central representability question for Shimura varieties beyond the cases covered by the established theory; the supplied status is unknown.