Fargues–Rapoport conjecture on admissibility and weak admissibility

Assume that $F$ has characteristic $0$ and that $\mu$ is minuscule. Let $G$ be the reductive group, let $b_0$ be the basic element associated with $(G,\{\mu\})$, and let ${\mathcal F}^{\rm a}(G,b_0,\{\mu\})$ and ${\mathcal F}^{\rm wa}(G,b_0,\{\mu\})$ denote the admissible and weakly admissible loci. The conditions in the preceding theorem are the equivalent conditions of full Hodge–Newton decomposability, minuteness of $\mu$, the zero-dimensionality condition for non-basic classes, the uniqueness condition for affine Deligne–Lusztig varieties, and the Deligne–Lusztig description of the basic locus. Fargues–Rapoport conjecture. These conditions are equivalent to

$${\mathcal F}^{\rm a}(G,b_0,\{\mu\})={\mathcal F}^{\rm wa}(G,b_0,\{\mu\}).$$

The conjecture gives a precise group-theoretic formulation of the expected relation between admissibility and weak admissibility; the source explains that this relation connects the period-domain question with Hodge–Newton decomposability and minute coweights.