Ichino–Ikeda–Harris refined Gan–Gross–Prasad conjecture for unitary groups

Let $F$ be a number field, let $E/F$ be a quadratic extension, and let $G$ and $H$ be the unitary groups in the source. Let $\pi=\pi_n\otimes\pi_{n+1}$ be a tempered cuspidal automorphic representation, let $\mathscr{P}$ be the automorphic period, and let $\alpha_v^{\natural}$ be the normalized local pairings. Write $S_\pi$ for the finite elementary $2$-group attached to the $L$-parameter of $\pi$. Assume that the global measure factors as $dh=\prod_v dh_v$. Ichino–Ikeda–Harris conjecture. For every decomposable vector $\phi=\otimes_v\phi_v\in\pi=\otimes_v\pi_v$,

$$\frac{|\mathscr{P}(\phi)|^2}{\langle\phi,\phi\rangle_{Pet}}=\frac{1}{|S_\pi|}\mathscr{L}\left(\frac{1}{2},\pi\right)\prod_v\frac{\alpha_v^{\natural}(\phi_v,\phi_v)}{\langle\phi_v,\phi_v\rangle_v}.$$

Here $\mathscr{L}(s,\pi)=\Delta_{n+1}L(s,\pi_E)/L(s+1/2,\pi,Ad)$. This is the refined form of the global Gan–Gross–Prasad conjecture: it predicts an exact period formula, including the component-group factor and normalized local contributions. The source attributes the conjecture to Ichino–Ikeda and N. Harris.