Igusa stacks product conjecture for minimal compactifications of infinite-level Shimura varieties

Establish the following two assertions for a Shimura datum (G,X) and compact open K^p \subseteq G(\mathbb{A}^{\infty p}): (1) Extend the Hodge–Tate period map to \pi_{HT}^{\diamond} : (Sh_{K^p}^{\ast})^{\diamond} \to Fl_{[\mu]}^{\diamond}. (2) Construct, compatibly in K^p, small v-stacks Igs_{K^p}^{\ast} with maps \overline{\pi}_{HT} : Igs_{K^p}^{\ast} \to Bun\,G and q_{K^p} : (Sh_{K^p}^{\ast})^{\diamond} \to Igs_{K^p}^{\ast} such that the canonical diagram with BL : Fl_{[\mu]}^{\diamond} \to Bun\,G is Cartesian.

Background

To construct inscribed global Shimura varieties, the authors invoke a fiber-product description involving the stack of G-bundles on the Fargues–Fontaine curve (Bun G) and the Hodge–Tate period domain Fl_{[\mu]}. This relies on a conjectural Igusa stacks factorization linking the minimal compactification of infinite-level Shimura varieties to Bun G via BL, together with an extension of the Hodge–Tate map.

This conjecture—proved in many PEL cases and compact Hodge type cases—would provide a Cartesian diagram realizing (Sh_{K^p}^{\ast})^{\diamond} as a fiber product of Fl_{[\mu]}^{\diamond} and an Igusa-type v-stack over Bun G, enabling the inscribed constructions and tangent computations presented in the paper.