Modularity conjecture for the arithmetic theta series

Let $S_\mathscr{L}$ be the space of functions used to define the coefficient classes $\widehat{\mathtt{Z}}^\mathrm{total}_\mathscr{L}(m)\in S_\mathscr{L}^\vee\otimes_\mathbb{C}\widehat{\mathrm{CH}}^1_\mathbb{C}(\mathtt{M}^*_\mathscr{L})$, and let $q=e^{2\pi i\tau}$. Define the formal generating series

$$\widehat{\theta}_\mathscr{L}(\tau)=\sum_{m\in\mathbb{Q}_{\geq 0}}\widehat{\mathtt{Z}}^\mathrm{total}_\mathscr{L}(m)q^m.$$

Modularity conjecture. The formal generating series is the $q$-expansion of a function

$$\widehat{\theta}_\mathscr{L}\in M_n(\omega_\mathscr{L}^\vee)\otimes_\mathbb{C}\widehat{\mathrm{CH}}^1_\mathbb{C}(\mathtt{M}^*_\mathscr{L}).$$

This predicts modularity of the generating series of arithmetic Kudla–Rapoport divisor classes, placing the classes in an automorphic generating function. The supplied source gives no resolution status.