Global distinction conjecture for unitary symmetric pairs

Let $F/F_0$ be a quadratic extension, let $F'$ be the quadratic étale algebra associated with the unitary symmetric pair, and let $\pi_0$ be a cuspidal automorphic representation of $\operatorname{GL}_{2n,F_0}$ of symplectic type. Assume that its base change ${\rm BC}_{F}(\pi_0)$ is cuspidal, and let $\pi$ be a cuspidal automorphic representation of the quasi-split unitary group $\operatorname{U}_{2n}$ of rank $2n$ over $F_0$ such that ${\rm BC}_{F}(\pi_0)={\rm BC}_{F}(\pi)$. Global distinction conjecture. The following two assertions are equivalent: one of $L(1/2,{\rm BC}_{F'}(\pi_0))$ and $L(1/2,{\rm BC}_{F'}(\pi_0\otimes\eta_{F/F_0}))$ does not vanish; there exists a unitary symmetric pair $(\operatorname{G},\mathrm{H})$ for an inner form of $\operatorname{U}_{2n}$ and an automorphic representation $\pi_{\operatorname{G}}$ on $\operatorname{G}(\mathbb A)$ nearly equivalent to $\pi$ such that $\pi_{\operatorname{G}}$ is $\mathrm{H}$-distinguished. This predicts that the relevant central $L$-value detects distinction across the appropriate inner forms, including unitary Friedberg--Jacquet and twisted variants; the source presents the paper as part of a program to establish cases of this conjecture.