Liu's refined Gan–Gross–Prasad conjecture for $({\rm SO}(5),{\rm SO}(2))$

Let $L/\mathbb Q$ be an imaginary quadratic extension with discriminant $-D$, let $\Lambda$ be a character of $L^\times\backslash\mathbb A_L^\times$ trivial on $\mathbb A^\times$, and let $\pi$ be an irreducible cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbb A)$ with trivial central character. Let $B(\phi,\Lambda)$ be the global Bessel period, let $J_v(\phi_v)$ be the normalized local factors, and suppose that $\pi_v$ is generic for almost all places $v$. For a factorizable automorphic form $\phi=\otimes_v\phi_v$ in the space of $\pi$, Liu's refined Gan–Gross–Prasad conjecture. One has

$$\frac{|B(\phi,\Lambda)|^2}{\langle\phi,\phi\rangle}=\frac{C_T}{S_\pi}\frac{\zeta_{\mathbb Q}(2)\zeta_{\mathbb Q}(4)L(1/2,\pi\times\operatorname{AI}(\Lambda^{-1}))}{L(1,\pi,\operatorname{Ad})L(1,\chi_{-D})}\prod_vJ_v(\phi_v),$$

where $\zeta_{\mathbb Q}(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$, $C_T$ is the constant relating the chosen local and global Haar measures, and $S_\pi$ is an integral power of $2$ related to the Arthur parameter of $\pi$, with

$$S_\pi=\begin{cases}4&\text{if $\pi$ is endoscopic,}\\2&\text{if $\pi$ is of general type.}\end{cases}$$

This is an explicit refined form of the Gan–Gross–Prasad conjecture for $({\rm SO}(5),{\rm SO}(2))$. The supplied context says that the underlying Gan–Gross–Prasad statement is now a theorem due to Furusawa and Morimoto, but gives no resolution evidence for this refined formula itself.