Global transfer conjecture for distinction of classical groups

Let $K/k$ be a quadratic extension of number fields, let $\mathbb{H}^\circ$ be a $k$-quasisplit classical group, and let $T$ be the functorial lift to ${\rm GL}_{m(n)}(\mathbb{A}_K)$. Let $\pi$ be an irreducible globally generic cuspidal automorphic representation of $\mathbb{H}^\circ(\mathbb{A}_K)$ such that $T(\pi)$ is cuspidal. Global distinction-transfer conjecture. (1) If $\pi$ is $\mathbb{H}^\circ(\mathbb{A}_k)$-distinguished, then $T(\pi)$ is ${\rm GL}_{m(n)}(\mathbb{A}_k)$-distinguished. (2) If $\pi$ is $(\mathbb{H}^\circ(\mathbb{A}_k),\omega_{\mathbb{H}^\circ,K/k})$-distinguished, then $T(\pi)$ is $({\rm GL}_{m(n)}(\mathbb{A}_k),\chi_{{\rm GL}_{m(n),K/k}})$-distinguished, where $\chi_{{\rm GL}_{m(n),K/k}}$ is the trivial character in the even orthogonal case and equals $\omega_{{\rm GL}_{m(n)},K/k}$ otherwise. This proposes a global analogue of the local distinction-transfer phenomenon; the stated functorial lift is known for globally generic cuspidal representations, but the asserted implications for distinction are left as conjectural.