On $\mathrm{S}\mathrm{p}$-distinguished representations of the quasi-split unitary groups

We study ${\mathrm{S}\mathrm{p}}_{2n}(F)$-distinction for representations of the quasi-split unitary group $U_{2n}(E\mathrm{/}F)$ in $2n$ variables with respect to a quadratic extension $E\mathrm{/}F$ of $p$-adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Moeglin-Tadic classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$-packets with no distinguished members that transfer under stable base change to ${\rm Sp}{2n}(E)$-distinguishedrepresentationsof${\rm GL}{2n}(E)$.