Perfectoidness conjecture for the spaces W_{n,d}

Let $n$ be a positive integer and, for each integer $d$ with $0<d<n$, let $W_{n,d}$ be the space defined in the paper from the corresponding local geometric construction. Perfectoidness conjecture. For all integers $0<d<n$, the space $W_{n,d}$ is perfectoid.

Perfectoidness of $W_{n,d}$ would extend the known cases $d=1$ and $d=n-1$, established using global Shimura varieties and local methods respectively. The conjecture concerns the remaining intermediate values of $d$.