Wan–Zhang weak global conjecture for the spherical variety $\mathbf{X}$

Let $F$ be a number field and let $G=\mathbf{GL}_4\times\mathbf{GL}_2$. Embed $\mathbf{GL}_2\times\mathbf{GL}_2$ into $G$ by $(g_1,g_2)\mapsto(\operatorname{diag}(g_1,g_2),g_2)$, and set

$$\mathbf{X}:=(\mathbf{GL}_2\times\mathbf{GL}_2)\setminus(\mathbf{GL}_4\times\mathbf{GL}_2).$$

Its associated $L$-group representation is

$$\rho_{\mathbf{X}}:=\wedge^2\otimes\mathrm{std}_2\oplus\mathrm{std}_4\oplus\mathrm{std}_4^\vee.$$

Let $\pi$ and $\sigma$ be cuspidal automorphic representations of $\mathbf{GL}_4$ and $\mathbf{GL}_2$, respectively, with trivial product of central characters. Wan–Zhang weak global conjecture. The following are equivalent:

1. $L(\frac{1}{2},\pi\otimes\sigma,\rho_{\mathbf{X}})\neq0$.
2. There exists a quaternion algebra $D/F$ such that $\mathcal{P}_{D^\times\times D^\times}|_{\pi_D\otimes\sigma_D}$ is not identically zero, where $\pi_D$ and $\sigma_D$ are related to $\pi$ and $\sigma$ by the Jacquet–Langlands correspondence.

Moreover, if these conditions hold, there exists a unique $D$ for which condition (2) holds. The conjecture is a weak global form of the expected relation between the central $L$-value and the period for this strongly tempered spherical variety; the source gives no evidence of a resolution.