Ichino–Ikeda type conjecture for strongly tempered spherical varieties

Let $(G,H)$ be one of the models in Table 1 other than the first, and let $D/k$ be a quaternion algebra, possibly split (or $D\in H^1(H/Z_{G,H},k)$ for Model 2). Let $\Pi_{\phi}=\bigcup_D\Pi_\phi(G_D)$ be a global tempered cuspidal $L$-packet with central character trivial on $Z_{G,H}(\mathbb A)$, let $\pi_D\in\Pi_\phi(G_D)$, and choose $\phi_D\in\nu(\pi_D)$. Let $S$ be a finite set of places outside which $\phi$ is unramified, and let $S_\phi$ be the conjectural global component group. The period integral $\mathcal P_{H_D}$ is taken with respect to the Tamagawa measure on $Z_{G_D,H_D}(\mathbb A)\backslash H_D(\mathbb A)$. Strong global Ichino–Ikeda conjecture. One has

$$|\mathcal P_{H_D}(\phi_D)|^2=\frac{1}{|S_{\phi}|}\cdot \frac{C_{H/Z_{G,H}}}{\Delta_{H_0/Z_{G,H}}(1)^S}\cdot \lim_{s\rightarrow 1} \frac{\Delta_G(s)^S}{L(1,\Pi_{\phi},\operatorname{Ad})^S}\cdot L(1/2,\Pi_{\phi},\rho_X)^S \cdot\prod_{v\in S} I_{H_{D,v}}(\phi_{D,v}).$$

Here the factors involving $\Delta$ and $L$ are partial $L$-functions, and $C_{H/Z_{G,H}}$ is the Haar measure constant of $H/Z_{G,H}$. This is the predicted global period formula for the models under consideration; it relates nonvanishing of periods to the central value of the relevant $L$-function, but its general validity remains conjectural.