The genericity conjecture for pure ABV-packets

Let $G$ be a reductive group and let $\phi$ be a Langlands parameter for $G$. Denote by $\Pi^{\text{pure}}_\phi(G)$ the associated pure $L$-packet, and call $\phi$ an open parameter when it is open in the parameter space defined by the paper. A representation is generic if it has a Whittaker model, and the packet is generic if it contains a generic representation.

Genericity conjecture. A pure $L$-packet $\Pi^{\text{pure}}_\phi(G)$ is generic if and only if $\phi$ is an open parameter for $G$.

This is the geometric characterization equivalent, in the paper's setting, to the Gross–Prasad–Rallis conjecture that genericity is equivalent to regularity at $s=1$ of the adjoint $L$-function. It is known for classical groups, including general linear groups, and in further cases under hypotheses on the local Langlands correspondence; the general statement is not established.