The Hecke-algebra structure conjecture for Bernstein components

Let $G$ be a reductive $p$-adic group, let $\mathfrak{s}=[L,\omega]_G$ be an inertial equivalence class, let $R_{\mathfrak{s}}$ be the root system defined by the poles of the Harish--Chandra $\mu$-function, let ${\mathfrak R}_{\mathfrak{s}}$ be the stabilizer of a positive Weyl chamber, and let $X_{\mathrm{nr}}(L)$ be the group of unramified characters of $L$. Hecke-algebra structure conjecture. There exist a parameter function $q_{\mathfrak{s}}:R_{\mathfrak{s}}\to\mathbb R_{>0}$, a finite-dimensional projective representation $V_{\mathfrak{s}}$ of $X_{\mathrm{nr}}(L)\rtimes{\mathfrak R}_{\mathfrak{s}}$, and a nice idempotent $e_{\mathfrak{s}}$ for $\mathfrak{s}$ such that

$$e_{\mathfrak{s}}\mathcal H(G)e_{\mathfrak{s}}\cong\Big(\mathcal H(X_{\mathrm{nr}}(L),R_{\mathfrak{s}},q_{\mathfrak{s}})\otimes\operatorname{End}_{\mathbb C}(V_{\mathfrak{s}})\Big)^{X_{\mathrm{nr}}(L)}\rtimes{\mathfrak R}_{\mathfrak{s}}.$$

This predicts the general Morita-type structure of Bernstein Hecke algebras, allowing nontrivial projective actions and cocycles; it is stated on the basis of known examples and remains open in general.