Scholze's fiber product conjecture for Igusa stacks

Let $(G,X)$ be a Shimura datum with associated $G(\mathbb{C})$-conjugacy class of minuscule cocharacters $[\mu^{-1}]$ and reflex field $E_0$. Fix a prime $p$, let $E$ be the completion of $E_0$ at the prime induced by an identification $\mathbb{C}\cong\overline{\mathbb{Q}}_p$, and write $\mathbb{F}_q$ for its residue field. For a compact open subgroup $K=K_pK^p\subseteq G(\mathbb{A}_f)$, let $\operatorname{Perf}$ be the v-site of perfectoid spaces in characteristic $p$, let $\mathcal{S}_{K^p}$ be the Shimura variety with infinite level at $p$, and let $\mathcal{S}_{K^p}/\operatorname{Spd}E$ denote its base change. Let $\operatorname{Gr}_G$ be the $B_{\operatorname{dR}}^+$-affine Grassmannian for $G_{\mathbb{Q}_p}$, let $\mu\in X_*^+(T)$ represent the conjugacy class of $[\mu]$, and let $\operatorname{Gr}_{G,\mu}$ be the corresponding Schubert cell. Let $\operatorname{Bun}_G$ be the v-stack of $G_{\mathbb{Q}_p}$-bundles on the Fargues–Fontaine curve, with Beauville–Laszlo uniformization map $BL:\operatorname{Gr}_G\to\operatorname{Bun}_G$ and Hodge–Tate period map $\pi_{HT}:\mathcal{S}_{K^p}\to\operatorname{Gr}_{G,\mu}$. Scholze's fiber product conjecture. There exists a construction of a system of small v-stacks $\{\operatorname{Igs}_{K^p}\}_{K^p}$ on $\operatorname{Perf}$, together with maps $\operatorname{red}:\mathcal{S}_{K^p}/\operatorname{Spd}E\to\operatorname{Igs}_{K^p}$ and $\overline{\pi}_{HT}:\operatorname{Igs}_{K^p}\to\operatorname{Bun}_G$, such that, for each $K^p$, the diagram with corners $\mathcal{S}_{K^p}$, $\operatorname{Gr}_{G,\mu}$, $\operatorname{Igs}_{K^p}$, and $\operatorname{Bun}_G$, and arrows $\pi_{HT}$, $\operatorname{red}$, $BL$, and $\overline{\pi}_{HT}$, is Cartesian; moreover, there exists a $G(\mathbb{A}_f)$-action on the system $\{\operatorname{Igs}_{K^p}\}_{K^p}$, with $G(\mathbb{Q}_p)$ acting trivially, that descends the Hecke action on $\{\mathcal{S}_{K^p}\}_{K^p}$. This conjecture seeks to separate the $p$-adic and prime-to-$p$ geometry of Shimura varieties at infinite level, with the latter encoded by Igusa stacks. Its resolution status is not established by the supplied source context.