Flicker–Rallis base-change conjecture for distinguished representations

Let $F$ be a $p$-adic field, let $E/F$ be a degree-two extension, set $G={\rm GL}_n(E)$ and $H={\rm GL}_n(F)$, and let $\pi$ be an irreducible admissible representation of $G$. A representation is $H$-distinguished if it admits a nonzero $H$-invariant linear form. The stable and unstable base-change maps from ${\rm U}(n,E/F)$ to ${\rm GL}_n(E)$ are understood in the usual sense. Flicker–Rallis conjecture. The representation $\pi$ is $H$-distinguished if and only if it is an unstable base-change lift from ${\rm U}(n)$ when $n$ is even, and a stable base-change lift from ${\rm U}(n)$ when $n$ is odd. This conjecture is known for $n=2$, and for $n=3$ when $\pi$ is in the discrete series; it remains open in general.