The Bernstein twisted extended-quotient conjecture

Let $F$) be a non-archimedean local field and $G$ a connected reductive algebraic group over $F$. For a Levi subgroup $L$ of $G$, write $W(G,L)=N_G(L)/L$. For an inertial equivalence class ${\frak s}=[L,\omega]_G$, let ${\rm Irr}^{\mathfrak s}(G)$ be the corresponding Bernstein component, let $T_{\mathfrak s}$ be its Bernstein torus, let $T_{{\mathfrak s},{\rm un}}$ be the unitary subtorus, and let $W_{\mathfrak s}$ be the finite stabilizer of $T_{\mathfrak s}$. For a finite group action, $T_{\mathfrak s}/\!/W_{\mathfrak s}$ denotes the extended quotient, with a family of twisting 2-cocycles $\natural$. Bijection with extended quotients. There exist a family of 2-cocycles $\natural$ and a bijection

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

such that its restriction is a bijection

$${\rm Irr}^{\mathfrak s}_{\rm temp}(G)\longleftrightarrow (T_{{\mathfrak s},{\rm un}}/\!/W_{\mathfrak s})_{\natural},$$

and this restriction determines the full bijection; moreover, if $\pi$ maps to $[t,\rho]$, then $W_{\mathfrak s}t$ is the unitary part of the cuspidal support of $\pi$ under the polar decomposition of $T_{\mathfrak s}$. This conjecture proposes a precise parametrization of each Bernstein component by a twisted extended quotient, with tempered representations corresponding to the unitary locus; its status is not resolved in the supplied source.