Kudla's generating-series conjecture for special cycles

Let $V$ be the quadratic space and $L$ an even lattice considered above, let $X(\Gamma_L)$ be the associated orthogonal Shimura variety, and let $S_{L,r}$ be the space of functions on $(L'/L)^r$ carrying the Weil representation $\omega_{L,r}$. For each positive semidefinite symmetric matrix $T\in\mathbb{Q}^{r\times r}$, let $Z(T)$ denote the corresponding element of $\operatorname{Hom}(S_{L,r},\operatorname{CH}^{r}(X(\Gamma_L))_\mathbb{C})$, obtained from the special cycle $Z(T,\varphi)$ by intersecting with $(\mathcal{L}^\vee)^{r-r(T)}$. Put

$$q^T=e^{2\pi i\operatorname{tr}(TZ)}\quad\text{for }Z\in\mathbb{H}_r.$$

Kudla's conjecture. The formal generating series

$$A_r(Z)=\sum_{\substack{T\in\mathbb{Q}^{r\times r}\\ T\geq 0}} Z(T)\cdot q^T$$

valued in $S_{L,r}^\vee\otimes_\mathbb{C}\operatorname{CH}^{r}(X(\Gamma_L))_\mathbb{C}$, is a Siegel modular form of genus $r$ in $M_{1+n/2}^{(r)}(\omega_{L,r}^\vee)$ with values in $\operatorname{CH}^{r}(X(\Gamma_L))_\mathbb{C}$. This is a Chow-valued form of the Kudla program, relating generating series of special cycles to Siegel modular forms; the supplied passage gives no resolution status, so the conjecture is retained as open.