Lapid–Mao's Ichino–Ikeda-type conjecture for Whittaker coefficients on the metaplectic group

Let $F$ be a number field, let $S$ be a sufficiently large finite set of its places, and let $\tilde\pi$ be the $\psi_{\tilde N}$-descent of an automorphic representation $\pi=\pi_1\boxplus\cdots\boxplus\pi_k$ of $\operatorname{GL}_{2n}(\mathbb A)$ as described above. For $\tilde\varphi\in\tilde\pi$ and $\tilde\varphi^{\vee}\in\tilde\pi^{\vee}$, write \tilde\mathcal W^{\psi_{\tilde N}} for the corresponding Whittaker coefficients, $N'$ for the subgroup appearing in the descent construction, and $\zeta_F^S$ and $L^S$ for the indicated partial zeta and $L$-functions. Lapid–Mao's conjecture. One has

\tilde\mathcal W^{\psi_{\tilde N}}(\tilde\varphi)\tilde\mathcal W^{\psi_{\tilde N}^{-1}}(\tilde\varphi^{\vee})=2^{-k}\left(\prod_{i=1}^n\zeta_F^S(2i)\right)\frac{L^S(\frac12,\pi)}{L^S(1,\pi,\operatorname{sym}^2)}\left(\operatorname{vol}(N'(\mathcal O_S)\backslash N'(F_S))\right)^{-1} $$\times\int^{st}_{N'(F_S)}(\tilde\pi(u)\tilde\varphi,\tilde\varphi^{\vee})_{\operatorname{Sp}_n(F)\backslash\operatorname{Sp}_n(\mathbb A)}\psi_{\tilde N}(u)^{-1}\,du.$$

Here $\mathcal O_S$ is the ring of $S$-integers, and $L^S(s,\pi,\operatorname{sym}^2)$ is the symmetric-square partial $L$-function. This is an analogue of the Ichino–Ikeda formula for Whittaker coefficients of generic representations on the metaplectic group; the source states the formula as a conjecture, and no resolution is supplied here.