Greer–Lian's modularity conjecture for special cycles on Siegel modular varieties

Let $\mathcal A_n=\operatorname{Sp}_{2n}(\mathbb Z)\backslash\mathbb H_n$ be the Siegel modular variety, and let $\lambda_i=c_i(\mathbb E)$ be the Chern classes of the Hodge bundle. For $m\geq1$, let $\mathcal Z(m)$ be the special cycle of principally polarized abelian varieties admitting an elliptic curve as an isogeny factor, with the multiplicities described in the source, and define

$$\Theta_{\mathcal A_n}(q)=\frac{(-1)^n}{24}\lambda_{n-1}+\sum_{m\geq1}[\mathcal Z(m)]q^m\in\operatorname{CH}^{n-1}(\mathcal A_n)\llbracket q\rrbracket.$$

Greer–Lian's modularity conjecture. The series $\Theta_{\mathcal A_n}(q)$ is a holomorphic modular form of weight $2n$ for $\operatorname{SL}_2(\mathbb Z)$, valued in $\operatorname{CH}^{n-1}(\mathcal A_n)$. The conjecture proposes Chow-valued modularity for the elliptic-factor special cycles on Siegel modular varieties.