Prasad's conjecture on distinction and base change for quasi-split classical groups

Let ${\bf Y}$ be an $F$-quasi-split group, let $E/F$ be the quadratic extension implicit in the notation, and let $\mathbf{BC}_F^E$ denote the base-change map from $L$-parameters of ${\bf Y}^{\mathrm{op}}(F)$ to those of ${\bf Y}(E)$. A representation of ${\bf Y}(E)$ is $E/F$-generic if it is generic with respect to a $\sigma$-stable maximal unipotent subgroup and a non-degenerate character satisfying $\psi^\sigma=\psi^{-1}$. Assume that the Langlands correspondence is known for ${\bf Y}(E)$ and ${\bf Y}^{\mathrm{op}}(F)$. Prasad's conjecture. (1) If $\pi$ is a $({\bf Y}(F),\omega_{{\bf Y},E/F})$-distinguished irreducible representation of ${\bf Y}(E)$, then the parameter of its $L$-packet belongs to $\mathbf{BC}_F^E(\Phi({\bf Y}^{\mathrm{op}}(F)))$. (2) Conversely, if the $L$-packet of an $E/F$-generic representation $\pi$ of ${\bf Y}(E)$ corresponds to a parameter in $\mathbf{BC}_F^E(\Phi({\bf Y}^{\mathrm{op}}(F)))$, then $\pi$ is $({\bf Y}(F),\omega_{{\bf Y},E/F})$-distinguished. This predicts a precise relationship between distinction and restriction of Langlands parameters, but its validity depends on the relevant cases of the local Langlands correspondence and remains unresolved in this generality.