Adler–Mishra–Stevens conjecture on the Bernstein-block local Langlands correspondence

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

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

Adler–Mishra–Stevens conjecture. The local Langlands correspondence for $L$ induces a bijection

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

This conjecture asserts compatibility of the local Langlands correspondence with Bernstein decompositions, matching the irreducible representations in the block indexed by $\mathfrak{s}$ with the enhanced $L$-parameters in the corresponding dual block. Its resolution is not established by the supplied text.