Special-case global Gan–Gross–Prasad nonvanishing conjecture

Let $F$ be a number field, let $S$ be an anisotropic symmetric $2\times2$ matrix over $F$, and let $T_S$ and $N$ be the associated torus and unipotent subgroup defining the Bessel period $B(\phi,\Lambda)$. Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GSp}(4,\mathbb A)$ with trivial central character, and let $\Lambda$ be a character of $T_S(F)\backslash T_S(\mathbb A)$ trivial on $\mathbb A_F^\times$. Assume that $\pi_v$ is generic for almost all places $v$.

Special case of global Gan–Gross–Prasad. If there is an automorphic form $\phi$ in the space of $\pi$ such that $B(\phi,\Lambda)\neq0$, then

$$L(1/2,\pi\times {\mathcal{AI}}(\Lambda^{-1}))\neq0.$$

This is the nonvanishing direction of the Bessel-period form of the Gan–Gross–Prasad conjecture. The source gives no resolution status.