Ichino–Ikeda type conjecture for the general linear quaternionic model

Consider the first model in Table 1, $(G,H)=({\mathrm{GL}}_4\times{\mathrm{GL}}_2,{\mathrm{GL}}_2\times{\mathrm{GL}}_2)$, for which $Z_H/Z_{G,H}\cong {\mathrm{GL}}_1$. Under the notation above, let $D/k$ be a quaternion algebra, let $\pi_D\in\Pi_\phi(G_D)$, and choose $\phi_D\in\nu(\pi_D)$. Quaternionic strong global Ichino–Ikeda conjecture. One has

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

The extra factor $\zeta_v(1)$ reflects the quotient $Z_H/Z_{G,H}\cong {\mathrm{GL}}_1$. This is the corresponding period formula for the first model, and its general validity remains conjectural.