Rapoport-Viehmann's local Shimura variety conjecture

Let $(G,[b],\{\mu\})$ be a local Shimura datum over $\mathbb{Q}_p$, with local reflex field $E$, flag variety $\mathscr{F}\ell_{G,\mu}$ over $\breve{E}$, weakly admissible locus $\mathscr{F}\ell_{G,\mu}^{wa}$, and associated group $J_b$. Rapoport–Viehmann's conjecture. There is a tower of rigid analytic spaces $(\mathcal{M}_K)_K$ over $\operatorname{Sp}\breve{E}$, indexed by the open compact subgroups $K$ of $G(\mathbb{Q}_p)$, such that $J_b(\mathbb{Q}_p)$ acts on each $\mathcal{M}_K$, $G(\mathbb{Q}_p)$ acts on the tower by Hecke correspondences, the tower has a Weil descent datum over $E$, and there is a compatible system of étale and partially proper period maps

that are equivariant for $J_b(\mathbb{Q}_p)$. This conjecture predicts the existence and basic structure of local Shimura varieties, including their symmetries, descent, and period maps. The source further notes a more precise expected description of the image of the period maps, but that refinement is separate from the stated tower conjecture.