Refined Gan–Gross–Prasad conjecture for Fourier–Jacobi periods

Let $F$ be a totally real number field, let $n'=n$ or $n'=n-1$, and let $\pi=\otimes'_v\pi_v$ and $\pi'=\otimes'_v\pi'_v$ be irreducible cuspidal automorphic representations with tempered $A$-parameters $\Psi$ and $\Psi'$, exactly one of which is genuine. Fix a sufficiently large finite set $S$ of bad places. Refined Gan–Gross–Prasad conjecture. For every place $v$, the local Hom-space

$$\operatorname{Hom}_{\operatorname{Sp}_{n'}(F_v)}(\pi_v\otimes\pi'_v\otimes\overline{\omega_{\psi_{\xi,v}}},\mathbb{C})$$

is nonzero if and only if the corresponding local period factor $\alpha_v$ is nonzero for suitable local vectors. Moreover, for factorizable vectors there is an explicit constant $\Delta^S$ such that

$$|\mathcal{P}_{n,n',\psi_\xi}(\varphi,\varphi',\phi)|^2 =\frac{2\Delta^S}{|A_\Psi||A_{\Psi'}|} \left.\frac{L^S(s,\pi\times\pi'\times\chi_\xi)}{L^S(s+1/2,\pi,\operatorname{Ad})L^S(s+1/2,\pi',\operatorname{Ad})}\right|_{s=1/2} \prod_{v\in S}\alpha_v(\varphi_v,\varphi'_v,\phi_v).$$

This refines the global Gan–Gross–Prasad conjecture by predicting an exact factorization of the squared global period into an $L$-value, adjoint factors, and local periods. The source presents it as conjectural; the local integral's absolute convergence is known.