Dijols–Prasad conjecture on Arthur packets and stable base change distinction

Let $E/F$ be a quadratic extension of $p$-adic fields, let $U_{2n}(E/F)$ be the quasi-split unitary group in $2n$ variables, and let $\operatorname{Sp}_{2n}(F)\subseteq U_{2n}(E/F)$ be the symplectic subgroup. An Arthur-type $L$-packet is an $L$-packet of irreducible representations of $U_{2n}(E/F)$ associated with an Arthur parameter. Its stable base change is the corresponding representation of $\operatorname{GL}_{2n}(E)$. A representation is symplectic-distinguished if it admits a nonzero invariant linear functional under the relevant symplectic group. Dijols–Prasad's conjecture. An Arthur-type $L$-packet of $U_{2n}(E/F)$ contains an $\operatorname{Sp}_{2n}(F)$-distinguished member if and only if its stable base change is an irreducible $\operatorname{Sp}_{2n}(E)$-distinguished representation of $\operatorname{GL}_{2n}(E)$. The paper studies this conjecture and proves related results, including cases where stable base change is distinguished but the relevant packet has no distinguished member; the conjecture itself is not stated as resolved.