Scholze's representability conjecture for integral local Shimura varieties

Let $(G,b,\mu)$ be a local Shimura datum, with $G$ a reductive group over $\mathbb{Q}_p$, $b\in G(\breve{\mathbb{Q}}_p)$, and $\mu$ a minuscule conjugacy class of cocharacters, and let $\mathcal{G}$ be a quasi-parahoric group scheme for $G$. Scholze's functor $\mathcal{M}^{\rm \int}_{\mathcal{G},b,\mu}$ is a $v$-sheaf over $\operatorname{Spd}(O_{\breve E})$. Scholze's conjecture. There exists a formal scheme $\mathscr{M}_{\mathcal{G},b,\mu}$, normal and flat and locally formally of finite type over $O_{\breve E}$, such that

$$\mathcal{M}^{\rm \int}_{\mathcal{G},b,\mu}=\mathscr{M}_{\mathcal{G},b,\mu}^{\Diamond}$$

as $v$-sheaves over $\operatorname{Spd}(O_{\breve E})$; moreover, this formal scheme is unique. This conjecture characterizes integral local Shimura varieties as formal models of Scholze's integral shtuka functors and is the main representability question addressed by the paper.