Theta-function coefficients in products of Jacobi forms

Let $L$, $K$, $v$, $w$, and $\iota$ be as in Theorem $\operatorname{MFprodJac}$. Write $D_L$ and $D_K$ for the discriminant groups of $L$ and $K$, and let $D_{L+_{\iota}K}$ be the discriminant group associated with the glued lattice. For $\gamma\in D_L$, $\beta\in D_K$, and $\alpha\in D_{L+_{\iota}K}$, let $\operatorname{p}_{\iota,\gamma,\beta}^{v,w,\alpha}(\tau)$ be the coefficient multiplying $\mathfrak{e}_{\alpha}^{*}$ in $\operatorname{P}_{\iota}^{v,w}(\mathfrak{e}_{\gamma}^{*}\otimes\mathfrak{e}_{\beta}^{*})$. Theta-function coefficient conjecture. There exists another lattice $\Lambda$ of the same signature, with an element $u\in\operatorname{Gr}(\Lambda_{\mathbb{R}})$, such that each coefficient $\operatorname{p}_{\iota,\gamma,\beta}^{v,w,\alpha}(\tau)$ is either zero or a theta function $\theta_{\Lambda+\delta}$ from Equation $\operatorname{Thetadef}$, for some $\delta\in[\operatorname{O}(\Lambda)\backslash D_{\Lambda}]$ depending on $\gamma$, $\beta$, and $\alpha$. This conjecture predicts that the coefficients in the product map are governed by theta functions attached to a lattice of the same signature, extending the explicit low-dimensional examples discussed immediately beforehand; the general assertion remains unproved in the supplied text.