Scholze's local-model conjecture

Let $(G,\{\mu\},K)$ be a local model triple, with $G$ split over a tame extension, and let ${\mathbb M}^{\mathrm{loc}}_K(G,\{\mu\})$ be the associated flat projective local model over $O_E$. Write ${\mathbb M}^{\mathrm{loc},\diamond}_K(G,\{\mu\})$ for its associated diamond and ${\rm Gr}_{{\mathcal G},{\rm Spd}(O_E)}$ for the Beilinson–Drinfeld affine Grassmannian over ${\rm Spd}(O_E)$. Scholze's local-model conjecture. For every such local model triple, ${\mathbb M}^{\mathrm{loc}}_K(G,\{\mu\})$ satisfies Scholze's conjecture: it has generic fiber $X_{\{\mu\}}$, reduced special fiber, and its associated diamond admits an equivariant closed immersion

$${\mathbb M}^{\mathrm{loc},\diamond}_K(G,\{\mu\})\hookrightarrow {\rm Gr}_{{\mathcal G},{\rm Spd}(O_E)}.$$

The conjecture is verified in the paper under additional hypotheses, including the abelian-type case reduced to general linear groups, but remains open in the full stated generality.