The Shimura-variety local Langlands action conjecture

Let $(G,X)$ and $(G',X')$ be Shimura data with $G'$ a prime-to-$p$ trivialized inner form of $G$, and let $K'_p\subset G'({\mathbb Q}_p)$ be open compact. Let $\Loc_{\widehat G,p}$ be the local parameter stack, let ${\mathfrak A}_{K_p,\Lambda}$ and ${\mathfrak A}_{K'_p,\Lambda}$ be the conjectural coherent sheaves attached to the corresponding local levels, and let $\widetilde{V_\mu}$ and $\widetilde{V_{\mu'}}$ be the sheaves associated with the relevant cocharacters. The Shimura-variety local Langlands action conjecture. For every specialization map $\operatorname{sp}:\overline\eta\to\overline v$, there is a natural map

$$R\operatorname{Hom}_{\Coh(\Loc_{\widehat G,p}\otimes\Lambda)}(\widetilde{V_\mu}\otimes {\mathfrak A}_{K_p,\Lambda},\widetilde{V_{\mu'}}\otimes {\mathfrak A}_{K'_p,\Lambda}) \to R\operatorname{Hom}_{H_{K^p,\Lambda}}(C_c(\Sh_K(G)_{\overline\eta},\Lambda[d_\mu]),C_c(\Sh_{K'}(G')_{\overline\eta},\Lambda[d_{\mu'}]))$$

compatible with compositions; in particular, the induced $E_1$-algebra map and the resulting $H_{K_p,\Lambda}$-action agree with the natural Hecke action and are independent of $\operatorname{sp}$. This conjecturally relates coherent local Langlands operations to morphisms between Shimura-variety cohomologies.