Guo's distinction transfer conjecture for general linear groups

Let $k$ be a number field with adele ring ${\mathbb A}$, let ${\mathbf G}={\operatorname{GL}}_{2n}$ and ${\mathbf H}={\operatorname{GL}}_n\times{\operatorname{GL}}_n$ embedded diagonally in ${\mathbf G}$, and let $k'/k$ be a quadratic extension with associated quadratic character $\eta$ of ${\mathbb A}^{\times}/k^{\times}$. Let ${\mathbf D}$ be a quaternion algebra over $k$ containing $k'$, set ${\mathbf G}'={\operatorname{GL}}_n({\mathbf D})$ and ${\mathbf H}'={\operatorname{GL}}_n(k')$, and suppose the representations below have trivial central character. Guo's distinction transfer conjecture. If there exists such $({\mathbf G}',{\mathbf H}')$ and an ${\mathbf H}'$-distinguished cuspidal automorphic representation $\pi'$ of ${\mathbf G}'({\mathbb A})$ which is the global Jacquet--Langlands transfer of a cuspidal automorphic representation $\pi$ of ${\mathbf G}({\mathbb A})$, then $\pi$ is both ${\mathbf H}$-distinguished and $({\mathbf H},\eta)$-distinguished. The conjecture predicts transfer of distinction from the quaternionic inner form to both relevant periods on the split group; the source also states that, when $n$ is odd, the converse should hold.