Multiplicity-one conjecture for distinguished representations of $\operatorname{GL}_{2n}(F)$

Let $E/F$ be a quadratic extension. Let $\pi$ be an irreducible representation of $\operatorname{GL}_{2n}(F)$, and let $\chi$ be a character of $E^{\times}$. Multiplicity-one conjecture. One should have

$$\dim_{\mathbb{C}}\operatorname*{Hom}_{\operatorname{GL}_n(E)}(\pi,\chi\circ\det)\leq 1.$$

Multiplicity one is known for cuspidal representations in the relevant distinguished-representation settings, but the source notes that it is unavailable for arbitrary irreducible representations; this conjecture proposes the bound for the subgroup $\operatorname{GL}_n(E)$.