Gross–Prasad conjecture on generic members and adjoint L-functions

Let $G$ be a split connected reductive group over a nonarchimedean local field $F$, let ${}^LG$ be its $L$-group with connected component $4\widehat{G}4$, and let $\varphi: W'_F\to {}^LG$ be an $L$-parameter. Write $\Pi_{\varphi}(G)$ for the associated $L$-packet, and let $L(s,\operatorname{Ad}_{\widehat{G}}\circ\varphi)$ be the adjoint $L$-function attached to the adjoint action of $\widehat{G}$ on its Lie algebra. A member of $\Pi_{\varphi}(G)$ is generic if it admits a Whittaker model. Gross–Prasad's conjecture. $\Pi_{\varphi}(G)$ contains a generic member if and only if

$$L(s,\operatorname{Ad}_{\widehat{G}}\circ\varphi)$$

is regular at $s=1$. Equivalently, a representation $\pi$ is generic if and only if $L(s,\pi,\operatorname{Ad}_{\widehat{G}})$ is regular at $s=1$. This conjecture relates genericity within an $L$-packet to the analytic behavior of its adjoint $L$-function; the supplied source identifies it as a conjecture of D. Gross and D. Prasad, and does not state whether it has been resolved in this generality.