Refined Gan–Gross–Prasad conjecture for unitary and orthogonal groups

Let $\mathbb F$ be a number field, let $\mathbb E/\mathbb F$ be either trivial or quadratic, and let $\mathscr G(\mathcal V)$ and $\mathscr G(\mathcal W)$ be the groups attached to the relevant sesquilinear spaces. Let $\pi_\mathcal V$ and $\pi_\mathcal W$ be tempered cuspidal automorphic representations appearing with multiplicity one, let $S$ contain the archimedean and ramified places, and let $\mathcal P(\varphi,\varphi')$ be the global period and $\alpha_v(\varphi_v,\varphi'_v)$ the local Fourier transforms. Liu's refined Gan–Gross–Prasad conjecture. For decomposable smooth vectors, the local Fourier transforms satisfy the stated absolute-convergence and nonvanishing conditions, and

$$\left|\mathcal P(\varphi,\varphi')\right|^2= \frac{1}{|\mathcal S_{\pi_\mathcal V}|\,|\mathcal S_{\pi_\mathcal W}|} \frac{\Delta_{\mathscr G(\mathcal V)}L^S(\tfrac12,\pi_\mathcal V\boxtimes\pi_\mathcal W)}{L^S(1,\pi_\mathcal V,\operatorname{Ad})L^S(1,\pi_\mathcal W,\operatorname{Ad})} \prod_{v\in S}\alpha_v(\varphi_v,\varphi'_v).$$

This refines the Gan–Gross–Prasad prediction by relating the square of the global period to the central $L$-value, adjoint $L$-values, component-group factors, and local pairings. The supplied text presents it as conjectural and gives no resolution evidence.