Jacquet–Ye base-change conjecture for unitary distinguished representations

Let $F$ be a $p$-adic field, let $E/F$ be a degree-two extension, let ${\rm U}(n)$ be the corresponding unitary group, and let $\pi$ be a representation of ${\rm GL}_n(E)$. A representation is ${\rm U}(n)$-distinguished if it admits a nonzero ${\rm U}(n)$-invariant linear form. Jacquet–Ye conjecture. The representation $\pi$ is distinguished with respect to ${\rm U}(n)$ if and only if it is a base-change lift from ${\rm GL}_n(F)$. The conjecture is known globally for $n=2,3$, while the general statement remains open.