Rapoport–Zink duality conjecture for basic local Shimura data

Let $(G,[b],\{\mu\})$ be a basic local Shimura datum, and let $(G^\vee,[b^\vee],\{\mu^\vee\})$ be its dual datum, with $G^\vee=J_b$, $[b^\vee]=[b^{-1}]$, and $\{\mu^\vee\}=\{\mu^{-1}\}$. Let $F$ be the base local field. Duality conjecture. There exists an isomorphism in the inverse-limit towers

$$\varprojlim_K \mathbb{M}(G,[b],\{\mu\})^K\simeq \varprojlim_{K^\vee}\mathbb{M}(G^\vee,[b^\vee],\{\mu^\vee\})^{K^\vee},$$

compatible with the actions of $G^\vee(F)\times G(F)$ on both sides. This is the local-Shimura-variety analogue of the duality conjecture of Rapoport–Zink; the source presents it conjecturally and assumes the relevant limiting objects.