Jacquet's distinguishedness conjecture for representations of ${\rm GL}_n(E)$

Let $F$ be a $p$-adic field, let $E/F$ be a degree-two extension with nontrivial Galois element $\sigma$, set $G={\rm GL}_n(E)$ and $H={\rm GL}_n(F)$, and let $\omega_{E/F}$ be the quadratic character associated to $E/F$. For an irreducible admissible representation $\pi$ of $G$, write $\omega_\pi$ for its central character and call $\pi$ $H$-distinguished if it admits a nonzero $H$-invariant linear form; call it $\omega_{E/F}$-distinguished if it admits a nonzero $H$-equivariant linear form transforming by $\omega_{E/F}$. Jacquet's conjecture. Let $\pi$ be an irreducible admissible representation of ${\rm GL}_n(E)$ such that $\omega_\pi|_{F^*}=1$ and $\pi^\vee\cong\pi^\sigma$. Then $\pi$ is $H$-distinguished if $n$ is odd. If $n$ is even, $\pi$ is either distinguished or $\omega_{E/F}$-distinguished with respect to $H={\rm GL}_n(F)$. The conjecture is settled when $\pi$ is square integrable, but remains part of the general picture for arbitrary irreducible admissible representations.