Generalized crude Siegel–Weil conjecture for Speh theta representations

Let $\tau$ be as in the surrounding discussion, and let $\theta_{\Delta(\tau,\ell)}\in\Theta_{\Delta(\tau,\ell)}$ and $\theta_{\Delta(\tau,2n+(2d-1)\ell)}\in\Theta_{\Delta(\tau,2n+(2d-1)\ell)}$. Let $U$ and its character $\psi_U$ be the unipotent data described in the source, and let $f_{\Delta(\tau,2n),s}$ be a section. Generalized crude Siegel–Weil conjecture. If $1\leq\ell\leq n$, there is a section $f_{\Delta(\tau,2n),s}$ for which the theta integral is the value at $s=\ell-n$ of a holomorphic Eisenstein series; if $n<\ell\leq2n$, there is a section for which the theta integral is the residue at $s=\ell-n$:

$$\int_{\operatorname{Sp}_{4d\ell}(F)\backslash\operatorname{Sp}_{4d\ell}(\mathbb A)}\theta_{\Delta(\tau,2n+(2d-1)\ell)}^{\psi_U}((h,g))\theta_{\Delta(\tau,\ell)}(g)\,dg=\operatorname{Value}_{s=\ell-n}E(f_{\Delta(\tau,2n),s},h)$$

for $1\leq\ell\leq n$, and

$$\int_{\operatorname{Sp}_{4d\ell}(F)\backslash\operatorname{Sp}_{4d\ell}(\mathbb A)}\theta_{\Delta(\tau,2n+(2d-1)\ell)}^{\psi_U}((h,g))\theta_{\Delta(\tau,\ell)}(g)\,dg=\operatorname{Res}_{s=\ell-n}E(f_{\Delta(\tau,2n),s},h)$$

for $n<\ell\leq2n$.