The modularity conjecture for higher theta series

Let $n\geq 1$ and $n\geq m\geq 1$. For a triple $(\mathcal{G},h,\mathcal{E})$ in the moduli stack of skew-Hermitian bundles with a Lagrangian sub-bundle, let $\Theta^r(\mathcal{G},h,\mathcal{E})$ be the higher theta series defined from the virtual classes $[\mathrm{Z}^r_{\mathcal{E}}(a)]^{\operatorname{vir}}$. Modularity conjecture. The function $\Theta^r(\mathcal{G},h,\mathcal{E})$ is independent of the choice of Lagrangian sub-bundle $\mathcal{E}\subset\mathcal{G}$. This independence is the geometric expression of modularity for the higher theta series, predicted by analogy with Kudla's conjectures.