The Gross-Prasad--Rallis genericity conjecture

Let $G$ be a quasi-split connected reductive group over a non-Archimedean local field, and let $\phi$ be a local $L$-parameter of $G$. Call $\phi$ generic when it is associated with a generic representation, and call it open when the corresponding orbit in the parameter variety is open. Let $L(s,\phi,\operatorname{Ad})$ be the adjoint local $L$-function.

Gross-Prasad--Rallis conjecture. The parameter $\phi$ is generic if and only if

$$L(s,\phi,\operatorname{Ad})$$

is regular at $s=1$.

The paper states that this is equivalent to the generic-if-and-only-if-open formulation and records a proof by Gan and Ichino under hypotheses known for general linear and classical groups. Thus, in the setting covered by that result, the conjecture is solved.