Crude Siegel–Weil formula for theta correspondences

Let $1 leq \ell<n$. Let $\theta_{\Delta(\tau,\ell)}$ and $\theta_{\Delta(\tau,2n+\ell)}$ be given \theta functions, and let $f'_{\Delta(\tau,2n),s}$ be a section of the relevant induced representation. For $h\in \operatorname{Sp}_{8n}(\mathbb A)$, Crude Siegel–Weil formula. There is a choice of $f'_{\Delta(\tau,2n),s}$ such that

$$\int_{\operatorname{Sp}_{4\ell}(F)\backslash \operatorname{Sp}_{4\ell}(\mathbb A)}\theta_{\Delta(\tau,2n+\ell)}(g,h)\theta_{\Delta(\tau,\ell)}(g)\,dg=\operatorname{Res}_{s=n-\ell}E(f'_{\Delta(\tau,2n),s})(h).$$

This is proposed as a formal Siegel–Weil-type identity: the theta integral is expected to be represented by a residue of an Eisenstein series, although convergence and the precise choice of section remain to be established.