Finite intersection conjecture for special cycles and CM cycles

Let $L$ be the relevant lattice, let $U$ be the negative-definite subspace defining the CM cycle, and let $P$ and $N$ be the positive- and negative-definite lattices arising from a splitting $V=V_+\oplus U$. For $\mu\in L'/L$ and positive $m\in Q(\mu)+\mathbb{Z}$, let $\mathcal{Z}(m,\mu)$ and $\mathcal{Z}(U)$ be the corresponding arithmetic cycles. Write $r(m,\mu)$ for the $(m,\mu)$-th coefficient of $\theta_P$ and $\kappa(m,\mu)$ for the $(m,\mu)$-th coefficient of $\mathcal{E}_N$. Finite intersection conjecture. The finite intersection pairing satisfies

$$\left\langle \mathcal{Z}(m,\mu),\mathcal{Z}(U)\right\rangle_{fin}=-\frac{2}{\operatorname{vol}(K_T)}\sum_{\substack{\mu_1\in P'/P\\\\mu_2\in N'/N\\\\mu_1+\mu_2\equiv \mu\\;(L)}}\sum_{\substack{\\\\ m_i\in \mathbb{Q}_{\geq 0} \\ m_1+m_2=m}}r(m_1,\mu_1)\kappa(m_2,\mu_2).$$

Equivalently, it is $-2/\operatorname{vol}(K_T)$ times the $(m,\mu)$-th Fourier coefficient of $\theta_P\otimes\mathcal{E}_N$. This conjecture is proposed to describe the finite-place contribution to the arithmetic intersection pairing; the supplied text gives no resolution status.