Unconditional equivalence between prismatic higher displays and integral local Shimura varieties

Let $(G,\mu,b)$ be an integral Shimura datum. Let $\mathcal{B}^{\textup{qsyn}}_{(G,\mu,b)}$ be the stack of prismatic higher $(G,\mu)$-displays with a quasi-isogeny to the fixed display associated with $(G,\mu,b)$, and let \mathcal{M}^{\textup{\int}}(G,\mu,b) be the integral local Shimura variety. Write $\mathfrak{Fib}_G$ for the natural transformation from the former to the latter.

Unconditional local Shimura conjecture. The natural transformation

\mathfrak{Fib}_G:\mathcal{B}^{\textup{qsyn}}_{(G,\mu,b)}\to\mathcal{M}^{\textup{\int}}(G,\mu,b)

is an isomorphism.

The paper proves this under the assumption that the integral local Shimura variety is representable by the diamond of a formal scheme, and notes that the assumption is expected to hold but should be removable. The claim is known in the Hodge-type case, where representability is established.