Virtual modularity conjecture for higher-codimension elliptic-factor cycles

Let $1<g\leq n$, let $\mathcal Z(M)$ be the higher-codimension special cycles on $\mathcal A_n$ associated with positive definite matrices $M\in\operatorname{Sym}_{g\times g}(\mathbb Z)$, and define the virtual generating series

$$\Theta^{g,\mathrm{vir}}_{\mathcal A_n}(\tau)=\sum_{M\geq0}\operatorname{pr}_*\left[\lambda_1^{\binom g2}\cdot\operatorname{cl}\left(\widetilde{\mathcal Z}(M)\right)\right]q^M.$$

Virtual modularity conjecture. The series $\Theta^{2,\mathrm{vir}}_{\mathcal A_n}$ is a holomorphic Siegel modular form of weight $2n$ for $\operatorname{Sp}_4(\mathbb Z)$, valued in $\operatorname{CH}^{2n-1}(\mathcal A_n)$, and vanishes at all cusps. The virtual correction is motivated by excess intersections; the source notes that the corresponding coefficients are nonzero only for $g\leq2$.