Ginzburg–Rallis global conjecture for the exterior-cube L-function

Let $k$ be a global field with ring of adeles ${\boldsymbol A}$, and let $D$ range over quaternion algebras over $k$. For an irreducible cuspidal automorphic representation $\pi$ of $\operatorname{GL}_6({\mathbb A})$ with central character $\omega_\pi=\chi^2$ for an idele character $\chi$ of ${\mathbb A}^{\times}/k^{\times}$, let $\pi_D$ denote its Jacquet–Langlands transfer to $\operatorname{GL}_3(D)({\mathbb A})$ when it exists, and let ${\mathcal P}_{R_D,\sigma_D\otimes\xi_D}$ be the corresponding Ginzburg–Rallis period. Ginzburg–Rallis conjecture. The central value $L^S(\frac{1}{2},\pi,\Lambda^3)$ does not vanish if and only if there is a unique quaternion algebra $D$ over $k$ such that $\pi_D$ exists and ${\mathcal P}_{R_D,\sigma_D\otimes\xi_D}(\phi^D)\ne0$ for some $\phi^D\in\pi_D$, while ${\mathcal P}_{R_{D'},\sigma_{D'}\otimes\xi_{D'}}(\phi^{D'})$ vanishes identically for every quaternion algebra $D'$ not isomorphic to $D$ over $k$ and every $\phi^{D'}\in\pi_{D'}$. This is the global period–central-value conjecture for the Ginzburg–Rallis model, analogous to the global Gan–Gross–Prasad and Jacquet conjectures; the source notes that it is encompassed by the general conjectural framework for periods on spherical varieties, and does not indicate a resolution.