The Schwartz-algebra conjecture for Bernstein components

Let $F$ be a non-archimedean local field, $G$ a connected reductive algebraic group over $F$, and ${\mathfrak s}=[L,\omega]_G$ an inertial equivalence class. Let $X_{\rm nr}(L,\omega)$ be the finite subgroup of unramified characters fixing $\omega$, let $X_{\rm unr}(L)$ be the unitary unramified characters, let $W_{\mathfrak s}$ be the Bernstein stabilizer, and let ${\mathcal S}(G)^{\mathfrak s}$ be the corresponding ideal in the Harish-Chandra--Schwartz algebra. Schwartz algebras. There exist a projective representation $V_{\mathfrak s}$ of $X_{\rm nr}(L,\omega)\rtimes W_{\mathfrak s}$ and a homomorphism

$$\zeta_G^{\mathfrak s}:\big(C^\infty(X_{\rm unr}(L))\otimes\operatorname{End}_{\mathbb C}(V_{\mathfrak s})\big)^{X_{\rm nr}(L,\omega)}\rtimes W_{\mathfrak s}\longrightarrow {\mathcal S}(G)^{\mathfrak s}$$

satisfying the stated canonical bijections of irreducible representations, spectrum preservation with respect to filtrations, and compatibility of the induced tempered parametrization with the extended-quotient bijection. This conjecture seeks an explicit Schwartz-algebra model for each Bernstein component and its tempered dual; the supplied source gives no resolution status.