Kudla's boundary-correction conjecture for compactified Shimura varieties

Let $X_\Gamma^\Sigma$ be a toroidal compactification of an orthogonal type Shimura variety, or of a unitary type Shimura variety of signature $(n+1,1)$. The special cycles of codimension $g\geq 1$ have a generating series, and boundary corrections are allowed to modify this series. Kudla's boundary-correction conjecture. There exist boundary corrections to the generating series of special cycles of codimension $g\geq 1$ such that the resulting series is a holomorphic Siegel modular form in the orthogonal case, respectively a holomorphic Hermitian modular form in the unitary case, valued in $\mathrm{CH}^g(X_\Gamma^\Sigma)$. This conjecture predicts that the modularity of special-cycle generating series extends to toroidal compactifications after accounting for boundary contributions; the source states it as the conjectural framework motivating the paper, while the paper proves the cohomological version in the relevant range rather than resolving the Chow-group statement in full.