Refined twisted Gan–Gross–Prasad period formula

Assume that $\pi$ is a tempered cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb A_E)$. Let $\varphi=\otimes_v\varphi_v\in\pi$ and $\phi=\otimes_v\phi_v\in\mathcal S(\mathcal L(\mathbb A_F))$, and define the normalized local periods by

$$\alpha_v^\natural(\varphi_v,\phi_v,\varphi_v',\phi_v')=\frac{\alpha_v(\varphi_v,\phi_v,\varphi_v',\phi_v')}{\mathcal L(\frac12,\pi_v)}.$$

Refined twisted Gan–Gross–Prasad formula. One has

$$|\P(\varphi,\phi)|^2=\mathcal L\left(\frac12,\pi\right)\prod_v\alpha_v^\natural(\varphi_v,\phi_v,\varphi_v,\phi_v).$$

This is the refined global period formula, conditional in the surrounding discussion on the unramified local conjecture; the source does not state an unconditional resolution of that conjecture.