Bültel–Pappas representability conjecture for adjoint nilpotent displays

Let $(G,\lbrace \mu \rbrace,[b])$ be an unramified local Shimura datum over $\mathbb{Q}_p$, let $E$ be its local Shimura field, and let $\breve{E}$ have residue field $k$. Let $\operatorname{\mathcal{G}}$ be a reductive group scheme over $\mathbb{Z}_p$ with generic fibre $G$, and choose an integral minuscule cocharacter

$$\mu\colon \mathbb{G}_{m,\mathcal{O}_E}\rightarrow \operatorname{\mathcal{G}}_{\mathcal{O}_E}.$$

Assume that $[b]\in B(G,\mu)$ has a representative $b=u_0\mu(p)$ with $u_0\in \operatorname{\mathcal{G}}(W(k))$, and that the isocrystal obtained from $b$ through the adjoint representation has all slopes greater than $-1$. Let $\mathcal{M}^{\operatorname{BP}}$ be the moduli problem of adjoint nilpotent $\operatorname{\mathcal{G}}$-$\mu$-displays up to isogeny.

Bültel–Pappas conjecture. The functor

$$\mathcal{M}^{\operatorname{BP}}\colon \operatorname{Nilp}_{W}^{\operatorname{op}}\rightarrow \operatorname{Set}$$

is representable by a formal scheme that is locally formally of finite type and formally smooth over $\operatorname{Spf}(\mathcal{O}_{\breve{E}})$.

This conjecture concerns the representability and smoothness of the display-theoretic moduli problem attached to an unramified local Shimura datum. The source supplies no resolution status, so it is recorded as open.