The equivariant K-theory conjecture for Bernstein components

Let $G$ be a reductive $p$-adic group, let $\mathfrak{s}\in{{\mathfrak B}}(G)$ be a Bernstein component, let $T_{\mathfrak{s},\mathrm{un}}$ be its unitary torus, let $W_{\mathfrak{s}}$ be the associated finite group, and let $\natural$ denote the relevant cocycle. Bernstein-component K-theory conjecture. There exists a canonical isomorphism

$$K^*_{W_{\mathfrak{s}},\natural}(T_{{\mathfrak{s}},\mathrm{un}})\to K_*(C_r^*(G)^{\mathfrak{s}}).$$

This is the topological K-theory counterpart of the extended-quotient conjecture and gives a finer prediction than Baum--Connes for each Bernstein component; the source does not report a general proof.