Drinfeld's algebraicity conjecture for Shimurian Barsotti–Tate stacks

Fix $n\in\mathbb N$ and $G\in\operatorname{\bf Shim}_n$. Let $S\mapsto\operatorname{BT}_n^G(S)$ be the contravariant functor defined on derived $p$-adic formal schemes of finite type over $\operatorname{Spf}\mathbb Z_p$. For each $m\in\mathbb N$, restrict this functor to derived schemes over $\mathbb Z/p^m\mathbb Z$.

Drinfeld's algebraicity conjecture. This restriction is a quasicompact smooth algebraic stack over $\mathbb Z/p^m\mathbb Z$ with affine diagonal.

This conjecture asserts that the syntomification-based construction has the expected geometric finiteness and smoothness properties over every finite mod-$p$ level. The source gives no resolution.