Kudla–Rapoport arithmetic Siegel–Weil conjecture for nonsingular intersections

Let $\mathcal{X}$ be a suitable regular integral model of the relevant orthogonal or unitary Shimura variety, let $n=\dim\mathcal{X}$, and let $\mathcal{Z}(m_i)$ be integral models of special divisors. For a positive definite symmetric matrix $T\in\operatorname{Sym}_n(F)_{>0}$, or a positive definite hermitian matrix $T\in\mathrm{Herm}_n(F)_{>0}$ in the unitary case, with diagonal entries $m_1,\ldots,m_n$, let $\langle\mathcal{Z}(m_1),\ldots,\mathcal{Z}(m_n)\rangle_T$ denote the corresponding $T$-part of the arithmetic intersection number. Kudla–Rapoport conjecture. Up to a nonzero constant depending only on choices of measures,

$$\langle\mathcal{Z}(m_1),\ldots,\mathcal{Z}(m_n)\rangle_T=E_T'(\tau,0),$$

where $E_T'(\tau,0)$ is the $T$-th Fourier coefficient of the central derivative of a Siegel Eisenstein series on $\operatorname{Sp}(2n)$, respectively $\mathrm{U}(n,n)$. The formula is known for arbitrary $n$ in the unitary case and in weaker semi-global form in the orthogonal case; the full orthogonal statement remains open.