The ABPS conjecture for Bernstein components

Let $G$ be a reductive $p$-adic group, let $\mathfrak{s}=[L,\omega]_G$ be an inertial equivalence class, let $T_{\mathfrak{s}}$ be the corresponding complex torus, and let $W_{\mathfrak{s}}$ be its finite Weyl group. For $t\in T_{\mathfrak{s}}$, write $W_{\mathfrak{s},t}$ for the stabilizer of $t$. ABPS conjecture. There exists a family of 2-cocycles

$$\natural_t:W_{\mathfrak{s},t}\times W_{\mathfrak{s},t}\to\mathbb C^\times\qquad(t\in T_{\mathfrak{s}})$$

and a bijection

$$\operatorname{Irr}(G)^{\mathfrak{s}}\longleftrightarrow (T_{\mathfrak{s}}/\!/W_{\mathfrak{s}})_{\natural}$$

that restricts to a bijection between tempered representations and the unitary part of the extended quotient, and is canonical up to permutations within L-packets. This conjecture proposes a geometric description of Bernstein components in terms of their cuspidal supports; its general validity and the precise compatibility with local Langlands remain open.