The local model conjecture for parahoric local Shimura pairs

Let $(G,\{\mu\})$ be a local Shimura pair, let $K$ be a parahoric subgroup of $G(\mathbb Q_p)$, and let ${\mathcal G}$ be the corresponding parahoric group scheme over $\operatorname{Spec}(\mathbb Z_p)$, with $K={\mathcal G}(\mathbb Z_p)$. Write $E$ for the local reflex field, ${\mathcal O}_E$ for its ring of integers, and $k$ for the residue field. Let ${\rm Adm}_{\tilde K}(\{\mu\})$ denote the associated admissible set, and let $X_\mu$ be the generic-fiber Schubert variety.

The local model conjecture. There exists a projective and flat scheme

$${\rm M}_K(G,\{\mu\})$$

over

$${\rm Spec}({\mathcal O}_E)$$

which supports an action of ${\mathcal G}\otimes_{\mathbb Z_p}{\mathcal O}_E$ and is such that: (i) the generic fiber ${\rm M}_K(G,\{\mu\})\otimes_{{\mathcal O}_E}E$ is $G_E$-isomorphic to $X_\mu$; (ii) there is a $\tilde K$-equivariant bijection between ${\rm M}_K(G,\{\mu\})(k)$ and

$$\{g\tilde K\in G(L)/\tilde K\mid \tilde K g\tilde K\in {\rm Adm}_{\tilde K}(\{\mu\})\}\subset G(L)/\tilde K.$$

This is a foundational conjecture relating integral local models to affine Grassmannian Schubert strata. The source presents it as an expected construction; no resolution is supplied in the given text.