Kudla's modularity conjecture for generating series of special cycles

Let $(V,Q)$ be a quadratic space over $\mathbb{Q}$ of signature $(n,2)$, let $D^+$ be a connected component of the associated orthogonal hermitian symmetric domain, let $L\subset V$ be an even lattice with dual lattice $L'$, and let $\Gamma\subset {\mathrm{O}}(L)$ be a finite-index subgroup acting trivially on $L'/L$ and preserving $D^+$. Set $X_\Gamma=\Gamma\backslash D^+$, and for $1\leq g\leq n$ let $S_{L,g}$ be the space of functions $(L'/L)^g\to\mathbb{C}$, with the Weil representation $\omega_{L,g}$ of ${\mathrm{Mp}}_{2g}(\mathbb{Z})$. For every positive semidefinite $t\in\operatorname{Mat}^{\mathrm{T}}_g(\mathbb{Q})$, let $Z(t)\in\operatorname{Hom}(S_{L,g},\operatorname{CH}^{g}(X_\Gamma)_\mathbb{C})$ denote the corresponding special cycle class. Kudla's modularity conjecture. The formal generating series

$$A_g(\tau)=\sum_{\substack{t\in\operatorname{Mat}^{\mathrm{T}}_g(\mathbb{Q})\\ t\geq 0}}Z(t)q^t$$

with coefficients in $S_{L,g}^{\vee}\otimes_\mathbb{C}\operatorname{CH}^{g}(X_\Gamma)_\mathbb{C}$ is a Siegel modular form in $M^{(g)}_{1+n/2}(\omega_{L,g}^{\vee})$ with values in $\operatorname{CH}^{g}(X_\Gamma)_\mathbb{C}$. This conjecture predicts modularity of generating series of special cycles on orthogonal Shimura varieties; the statement is known in important low-genus cases, including genus $2$, but is not established in the stated generality.