Finite arithmetic intersection conjecture for special cycles

Let $L$ be a lattice with discriminant group $L'/L$, let $\bar\rho_L$ be its associated representation, and let $\mathcal{X}_K$ be a regular projective flat integral model over $\operatorname{Spec}\mathbb{Z}$. For $\mu\in L'/L$ and positive $m\in Q(\mu)+\mathbb{Z}$, let $\mathcal{Z}(m,\mu)$ and $\mathcal{Z}(W)$ be the specified extensions to $\mathcal{X}_K$ of the corresponding cycles, and let $C(W,K)$ and $\mathcal{E}(\tau,L)$ have the meanings established in the paper. Finite arithmetic intersection conjecture. The cycles $\mathcal{Z}(m,\mu)$ and $\mathcal{Z}(W)$ intersect properly, and

$$\left\langle \mathcal{Z}(m,\mu),\mathcal{Z}(W)\right\rangle_{\mathrm{fin}}=-\frac{1}{2}C(W,K)\,[\mathcal{E}(\tau,L)]_{m,\mu},$$

where $[\mathcal{E}(\tau,L)]_{m,\mu}$ denotes the $(m,\mu)$-th Fourier coefficient. This is one of the equivalent conjectures inspired by the arithmetic intersection formula; it predicts that finite arithmetic intersections of special cycles are governed by Fourier coefficients of an Eisenstein series, but the supplied text gives no resolution evidence.