The unitary-group and symplectic distinction conjecture

Let $F$ be a local field, let $E$ be the quadratic extension used for base change, and let $\{\pi\}$ be an $L$-packet of irreducible admissible representations of $\operatorname{U}(n,n)(F)$, assumed to be the $L$-packet associated to an Arthur packet on $\operatorname{U}(n,n)(F)$. A representation is distinguished by a subgroup if it admits a nonzero invariant linear form under that subgroup, and $BC(\pi)$ denotes the base change of $\pi$ to $\operatorname{GL}_{2n}(E)$.

Unitary-symplectic distinction conjecture. Some member of $\{\pi\}$ is distinguished by $\operatorname{Sp}_{2n}(F)$ if and only if $BC(\pi)$ is distinguished by $\operatorname{Sp}_{2n}(E)$.

The paper indicates a proof in the case of $\operatorname{U}(2,2)$. Together with the stated classification of symplectically distinguished representations of $\operatorname{GL}_{2n}(E)$, the conjecture would imply that no tempered representations of $\operatorname{U}(n,n)(F)$ are distinguished by $\operatorname{Sp}_{2n}(F)$.