The genericity conjecture for L-packets and open Vogan orbits

Assume the $p$-adic Kazhdan–Lusztig hypothesis. Let $\phi$ be a Langlands parameter, let $\lambda_\phi$ be its infinitesimal parameter, and let $C_\phi$ be the orbit associated to $\phi$ in the Vogan variety $V_{\lambda_\phi}$. A $G$-representation is generic if it admits a Whittaker model. Genericity–open-orbit conjecture. If the L-packet $\Pi_\phi(G)$ contains a generic representation, then $C_\phi$ is open. By the cited result, openness of $C_\phi$ is equivalent to regularity of $L(s,\phi,\operatorname{Ad})$ at $s=1$, so this gives the forward direction of Gross–Prasad's conjecture; the parser marks the statement resolved.