Weak global Ichino–Ikeda conjecture for strongly tempered spherical varieties

Let $\Pi_\phi=\bigcup_D\Pi_\phi(G_D)$ be the global tempered cuspidal $L$-packet and retain the notation for the models $(G_D,H_D)$, the embeddings $\nu$, and the period integrals $\mathcal P_{H_D}$. Weak global Ichino–Ikeda conjecture. The following are equivalent:

1. $L(1/2,\Pi_\phi,\rho_X)\ne0$.
2. There exists a quaternion algebra $D/k$ (or $D\in H^1(H/Z_{G,H},k)$ in the case of Model 2) such that $\mathcal P_{H_D}(\phi_D)\ne0$ for some $\phi_D\in\nu(\pi_D)$ and $\pi_D\in\Pi_\phi(G_D)$.

Moreover, when these conditions hold, there exist a unique $D$ and a unique $\pi_D\in\Pi_\phi(G_D)$ satisfying condition 2. This weak form is presented as a consequence of the strong global conjectures together with local multiplicity-one theorems for Vogan packets; its general validity therefore remains conjectural.