Gan–Gross–Prasad conjecture for relevant pure inner forms

Let $(G_n,G_m)$ be one of the following pairs of classical groups:

$$(G_n,G_m) \in \{(\operatorname{SO}(V_n),\operatorname{SO}(V_m)),(\operatorname{Sp}(W_n),\widetilde{\operatorname{Sp}}(W_m)),(\widetilde{\operatorname{Sp}}(W_n),\operatorname{Sp}(W_m))\}.$$

Let $M \times N$ be a discrete global $A$-parameter of $G_n \times G_m$, and let $\pi_1 \boxtimes \pi_2$ be an irreducible genuine discrete automorphic representation associated with this parameter.

Gan–Gross–Prasad conjecture. If $n-m$ is odd, a nonzero Bessel period $\mathcal{B}_{k,\psi}^{\varepsilon}$ on $\pi_1 \boxtimes \pi_2$ implies that $(M,N)$ is a relevant pair. For a relevant pair, some representation in $\Pi_{M \times N}^{\mathrm{rel}}$ has nonzero Bessel period if and only if

$$L(0,M,N)\neq 0.$$

If $n-m$ is even, a nonzero Fourier–Jacobi period $\mathcal{FJ}_{k,\psi}^{1}$ on $\pi_1 \boxtimes \pi_2 \boxtimes \nu_{\psi^{-1},W_m}^{\lambda}$ implies that $(M,N)$ is a relevant pair. For a relevant pair, some representation in $\Pi_{M \times N}^{\mathrm{rel}}$ has nonzero Fourier–Jacobi period if and only if

$$L(0,M,N)\neq 0.$$

The conjecture unifies the Bessel and Fourier–Jacobi cases and incorporates all relevant pure inner forms. In the orthogonal case these forms vary, whereas the symplectic groups and their double covers have unique pure inner forms up to isomorphism. The source gives no evidence resolving the assertions.