The restriction conjecture for local Langlands packets of disconnected groups

Let $B\to A$ be a map of finite groups acting on a connected reductive group $G$, and write $G^A=G\rtimes A$ and $G^B=G\rtimes B$. Let $\pi^A$ correspond to $(\phi,\rho^A)$ for $G^A_z(F)$ and let $\pi^B$ correspond to $(\phi',\rho^B)$ for $G^B_z(F)$. The restriction conjecture. The multiplicity of $\pi^B$ in the restriction of $\pi^A$ to $G^B_z(F)$ is zero unless $\phi$ and $\phi'$ are $G^A$-equivalent. After choosing equivalent representatives with $\phi=\phi'$, this multiplicity equals the multiplicity of $\rho^B$ in the restriction of $\rho^A$ to the corresponding component group. This predicts compatibility of the local correspondence with restriction along maps of component groups; it remains conjectural in the stated generality.