Ichino–Ikeda refined Gross–Prasad conjecture for special orthogonal groups

Let $G_n\subset G_{n+1}$ be the special orthogonal groups in the Gross–Prasad setting, and let $\pi_n$ and $\pi_{n+1}$ be the associated tempered cuspidal automorphic representations. Let $\mathcal{P}$ be the global $G_n(\mathbb{A}_F)\times G_n(\mathbb{A}_F)$-invariant functional and let $\mathcal{P}_v$ be the normalized local functionals. Let $\Delta_{G_{n+1}}$ be the global motive factor, and let $\psi_i$ be the conjectural $L$-parameter for $\pi_i$, with associated component group $S_{\psi_i}=\operatorname{Cent}_{\widehat{G_i}}(\operatorname{Im}(\psi_i))$. Ichino–Ikeda refined Gross–Prasad conjecture. If $\beta$ is an integer satisfying $2^\beta=|S_{\psi_{n+1}}|\,|S_{\psi_n}|$, then

$$\mathcal{P}(\phi,f)=\frac{\Delta_{G_{n+1}}}{2^\beta}\frac{L(1/2,\pi_n\boxtimes\pi_{n+1})}{L(1,\pi_n,\operatorname{Ad})L(1,\pi_{n+1},\operatorname{Ad})}\prod_v\mathcal{P}_v(\phi_v,f_v).$$

This conjecture specifies the exact proportionality constant between the global period pairing and the product of normalized local pairings. The source states unconditional results for $n=2$ and $n=3$, and for $n=4$ under the stated theta-lift hypothesis.