Aubert–Moussaoui–Solleveld Bernstein-block LLC conjecture

Let $G$ be a reductive group and let $\mathfrak{B}(G)$ denote its Bernstein spectrum. For a Bernstein component $\mathfrak{s}=[L,\sigma]_G\in\mathfrak{B}(G)$, suppose the local Langlands correspondence for $L$ assigns to $\sigma$ the enhanced parameter $(\varphi_\sigma,\rho_\sigma)$. Define the corresponding dual Bernstein component by

$$\mathfrak{s}^{\vee}=[L^{\vee},(\varphi_\sigma,\rho_\sigma)]_{G^{\vee}}.$$

Aubert–Moussaoui–Solleveld's Bernstein-block conjecture. The local Langlands correspondence for $L$ induces a bijection

$$\operatorname{Irr}^{\mathfrak{s}}(G)\xrightarrow{1\text{-}1}\Phi_{\mathrm{e}}^{\mathfrak{s}^{\vee}}(G).$$

This conjecture refines the local Langlands correspondence block by block. The supplied text records cases where the related compatibility property is known, but does not state that this Bernstein-block conjecture itself is resolved.