The global Shimura variety local-model diagram conjecture

Let ${\mathscr S}_{\sf K}$ be an integral model of a Shimura variety equipped with its $v$-sheaf theoretic local model diagram. A scheme-theoretic local model diagram is a diagram of $O_E$-schemes

$$\widetilde{{\mathscr S}}_{\sf K}\xrightarrow{\pi}{\mathscr S}_{\sf K},\qquad \widetilde{{\mathscr S}}_{\sf K}\xrightarrow{q}{\mathbb M}_{{\mathcal G},\mu},$$

where $\pi$ is a ${\mathcal G}$-torsor, $q$ is ${\mathcal G}$-equivariant and smooth, the generic fiber is given by the Borel embedding construction, and applying the diamond functor gives the $v$-sheaf local model diagram. Global local-model diagram conjecture. A scheme-theoretic local model diagram for ${\mathscr S}_{\sf K}$ exists. It is uniquely determined if it exists; the source states that it is known in the global Hodge type case under additional tameness hypotheses, but not in general.