The existence conjecture for integral models of Shimura varieties of abelian type

Fix a prime $p$, a Shimura datum $({\mathbb G},X)$ of abelian type, a prime of its reflex field above $p$, the corresponding local reflex field $E$, and a parahoric subgroup $K_p\subset {\mathbb G}(\mathbb Q_p)$. Let $G={\mathbb G}_{\mathbb Q_p}$, let $\{\mu\}$ be the associated local conjugacy class, and assume a local model ${\rm M}_{K_p}(G,\{\mu\})$ as above is given. Write ${\mathcal G}^c={\mathcal G}/{\mathcal Z}_s$, where ${\mathcal Z}_s$ is the Zariski closure of the central torus $Z_s\subset G$ in ${\mathcal G}$.

The integral-model conjecture. There is a scheme ${\mathscr S}_{K_p}({\mathbb G},X)$ over

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

with a right action of ${\mathbb G}({\mathbb A}_f^p)$ satisfying: (a) sufficiently small open compact subgroups $K^p$ act freely, and the quotients are finite-type ${\mathcal O}_E$-schemes extending the corresponding generic Shimura varieties, with ${\mathscr S}_{K_p}=\varprojlim_{K^p}{\mathscr S}_{K_pK^p}$; (b) for every discrete valuation ring $R\supset{\mathcal O}_E$ of mixed characteristic $(0,p)$, the map ${\mathscr S}_{K_p}(R)\to{\mathscr S}_{K_p}(R[1/p])$ is a bijection; and (c) there is a smooth morphism of stacks

$$\lambda:{\mathscr S}_{K_p}({\mathbb G},X)\to[({\mathcal G}^c\otimes_{{\mathbb Z}_p}{\mathcal O}_E)\backslash {\rm M}_{K_p}(G,\{\mu\})]$$

that is invariant under the prime-to-$p$ action and whose generic fiber is the canonical principal ${\mathbb G}^c_E$-bundle.

These properties formulate the expected integral model for parahoric level. The source gives no resolution of this candidate in the supplied material, although it notes that the construction is known in some cases and uniqueness is unclear in general.