The dual generic representation conjecture

Let $G$ be a quasi-split connected reductive group over a non-Archimedean local field. Let $pi$ be a generic representation of $G$, let $\phi_pi$ be its local $L$-parameter, and let $\lambda=\lambda_{\phi_pi}$. Let $V_\lambda$ be the variety attached to $\lambda$, and let $\phi_0$ be the local $L$-parameter corresponding to the zero orbit in $V_\lambda$. Write $\widehat{pi}$ for the Aubert--Zelevinsky dual.

Dual generic representation conjecture. One has

$$\widehat{pi} e Pi_{\phi_0}.$$

The conjecture is proposed as the expected form of compatibility between generic representations, Aubert--Zelevinsky duality, and the parameter attached to the zero orbit. Its general status is open.