Conjectural Whittaker formula for mixed special-cycle intersections

Let $0\leq h\leq 2n$, and let $x_1,\dots,x_{2n-h},y_1,\dots,y_h\in {\mathbb V}$ be the special homomorphisms defining cycles ${\mathcal Z}(x_i)$ and ${\mathcal Y}(y_j)$. Let $B$ be their $(2n)\times(2n)$ hermitian Gram matrix, let $W_{h,t}(B,0)$ be the local Whittaker functions attached to the relevant Siegel–Weil sections, let $\beta_i^h$ be the constants defined by the preceding linear relation, and set $A_n=\operatorname{diag}(1_n,\pi^{-1}1_n)$. Mixed special-cycle intersection conjecture. The arithmetic intersection number satisfies

$$\left\langle {\mathcal Z}(x_1),\dots,{\mathcal Z}(x_{2n-h}),{\mathcal Y}(y_1),\dots,{\mathcal Y}(y_h)\right\rangle=\frac{1}{W_{n,n}(A_n,0)}\left\{W'_{h,n}(B,0)-\sum_{0\leq i\leq n-1}\beta_i^hW_{h,i}(B,0)\right\}.$$

This extends the Kudla–Rapoport formula to arbitrary mixtures of ${\mathcal Z}$- and ${\mathcal Y}$-cycles. The source proves the case $n=1$, while the general formula is presented as a conjecture.