Vogan L-packet genericity equivariance conjecture

Let $G$ be a quasi-split connected reductive group over a local field $k$ of characteristic zero, let $G'$ be a pure inner form, and let $\Pi'_{\varphi}$ be the standard $L$-packet of $G'$ inside the Vogan $L$-packet $\widetilde{\Pi}_{\varphi}$. Choose a Whittaker-compatible parametrization $\rho\mapsto\pi_{\rho}$ of $\widetilde{\Pi}_{\varphi}$, and let $\zeta'_{\varphi}:G'_{\operatorname{ad}}(k)/p(G'(k))\to\widehat{R_{\varphi}}$ be the canonical map described in the text. Vogan L-packet genericity equivariance conjecture. For $g\in G'_{\operatorname{ad}}(k)$, if $\pi_{\rho}\in\Pi'_{\varphi}$, then

$$\pi_{\rho}\circ\operatorname{Ad}(g)=\pi_{g\cdot\rho},\qquad g\cdot\rho=\rho\otimes\zeta'_{\varphi}(g).$$

Moreover, $\pi_{\rho}$ is $\psi$-generic if and only if $\pi_{g\cdot\rho}$ is $g\cdot\psi$-generic. This is the pure-inner-form version of the expected compatibility between packet parametrizations, conjugation, and genericity; the supplied text gives no resolution status.