Lapid–Mao conjecture on Whittaker and $L^2$ normalizations for $\mathrm{GSp}_4$

Let $G=\mathrm{GSp}_4$, let $\pi=\otimes_v\pi_v$ be an irreducible, unitary, cuspidal, generic automorphic representation of $G(\mathbb A)$ with trivial central character, and let $\phi=\otimes_v\phi_v$ be a factorizable vector in its space. Let $g_0=(g_{0,v})_v\in G(\mathbb A)$, and let $S$ contain the archimedean place such that, for every $p\notin S$, $\pi_p$ and $\phi_p$ are unramified and $g_{0,p}\in K_p$. Lapid–Mao conjecture. Then

$$\frac{|W_\phi(g_0)|^2}{\langle\phi,\phi\rangle}=2^{-c}\frac{\zeta^{S}(2)\zeta^{(S)}(4)}{L^{S}(1,\pi,\operatorname{Ad})}\prod_{v\in S}J_0(g_{0,v}\cdot\phi_v),$$

where $J_0(g_{0,v}\cdot\phi_v)$ is the local normalized factor, $L^{S}(1,\pi,\operatorname{Ad})$ is the degree-$10$ adjoint $L$-function with the factors in $S$ omitted, and

$$c=\begin{cases}1&\text{if $\pi$ is of general type,}\\2&\text{if $\pi$ is endoscopic.}\end{cases}$$

This conjecture gives an explicit relation between arithmetic/Whittaker normalization and $L^2$ normalization of globally generic cuspidal automorphic forms. The supplied context attributes it to Lapid and Mao, but provides no evidence that it has been proved or disproved.