4 problems

Let $({\sf G},X)$ be an arbitrary Shimura datum and let ${\sf K}={\sf K}_p{\sf K}^p$, with ${\sf K}_p={\mathcal G}(\mathbb Z_p)$ for a connected parahoric model ${\mathcal G}$ of…

The integral-model conjecture. There is a scheme ${\mathscr S}_{K_p}({\mathbb G},X)$ over … with a right action of ${\mathbb G}({\mathbb A}_f^p)$ satisfying: (a) sufficiently small…

Reduction of the parahoric torsor conjecture. The ${\mathcal G}$-torsor $\pi^\circ$ should admit a reduction to a ${\mathcal G}^\circ$-torsor.

Rapoport's local model diagram conjecture. The local model ${\rm M}^{\rm loc}_{G,X}$ should carry an action of ${\mathcal G}^\circ$, and there should be a smooth morphism of stacks