Modularity conjecture for d-elliptic Noether–Lefschetz cycles

For integers $d,g\ge 1$, let $\widetilde{\mathsf{NL}}_{g,d}$ be the moduli space of morphisms of abelian varieties $h:E\to A$, where $E$ is an elliptic curve, $A$ is a principally polarized abelian variety of dimension $g$ with polarization $\Theta_A$, and $\deg(h^*\Theta_A)=d$. Let $[\widetilde{\mathsf{NL}}_{g,d}]\in \operatorname{CH}^{g-1}(\mathcal{A}_g)$ be the associated Noether–Lefschetz cycle class, and set $[\widetilde{\mathsf{NL}}_{g,0}]=\frac{1}{24}(-1)^g\lambda_{g-1}$. Modularity conjecture. The generating series

$$\sum_{d\ge 0}[\widetilde{\mathsf{NL}}_{g,d}]q^d \in \operatorname{CH}^{g-1}(\mathcal{A}_g)\otimes \mathbb{Q}[[q]]$$

is a cycle-valued modular form of weight $2g$. The conjecture is known after projection to the tautological ring, while the full Chow-valued modularity statement remains open.