The integral-model conjecture for Igusa stacks and Shimura varieties

Let $(\mathsf{G},\mathsf{X})$ admit an Igusa stack $\mathrm{Igs}^\circ(\mathsf{G},\mathsf{X})$ at $v$. Given a parahoric model $\mathcal{G}$ of $G$ with $K_p=\mathcal{G}(\mathbb{Z}_p)$, define $S$ by the fiber product

\begin{tikzcd} S \arrow{r}{\pi_{\mathrm{crys}}} \arrow{d} & \mathrm{Sht}_{\mathcal{G},\mu} \arrow{d}{\mathrm{BL}^{\circ}} \\ \mathrm{Igs}^\circ(\mathsf{G},\mathsf{X}) \arrow{r}{\overline{\pi}_{\mathrm{HT}}} & \mathrm{Bun}_{G,\mu^{-1}}. \end{tikzcd}

Integral-model conjecture. The composition

$$S\longrightarrow \mathrm{Sht}_{\mathcal{G},\mu}\longrightarrow \mathrm{Sht}_{\mathcal{G}^c,\mu^c}$$

is $\underline{\mathsf{G}(\mathbb{A}_f^p)}$-equivariantly isomorphic to the morphism

$$\mathscr{S}_{K_p}(\mathsf{G},\mathsf{X})^{\diamond}\longrightarrow \mathrm{Sht}_{\mathcal{G}^c,\mu^c}.$$

This identifies the fiber product formed from the crystalline and Hodge–Tate period maps with the conjectural canonical integral model after passage to the cuspidal quotient. The existence of the relevant integral model is known in the stated settings of abelian type for $p\geq 3$ and of Hodge type, but the asserted equivariant identification is the remaining conjectural assertion.