Gross–Kohnen–Zagier conjecture on Green's functions at CM points

Let $\Gamma=SL_2(\mathbb Z)$, let $G_s(z_1,z_2)$ be the Green's function, and let

$$G_s^m(z_1,z_2)=G_s(z_1,z_2)|T_m=-2\sum_{\substack{\gamma\in M_2(\mathbb Z)\det(\gamma)=m}}Q_{s-1}\left(1+\frac{|z_1-\gamma z_2|^2}{2\operatorname{Im}(z_1)\operatorname{Im}(\gamma z_2)}\right).$$

For a weakly holomorphic modular form $f=\sum_m c_f(m)q^m$ of weight $-2j$, define

$$G_{j+1,f}(z_1,z_2)=\sum_{m>0}c_f(-m)m^jG^m_{j+1}(z_1,z_2).$$

For a discriminant $d<0$, let ${\mathcal O}_d$ be the ring of integers of $\mathbb Q(\sqrt d)$ and let $H_d$ be its Hilbert class field. Gross–Kohnen–Zagier's conjecture. If $c_f(m)\in\mathbb Z$ for all $m<0$, $z_1$ is a $CM$ point of discriminant $d_1$, and $z_2$ is a $CM$ point of discriminant $d_2$ such that $(z_1,z_2)$ does not lie on $Z(f)=\sum c_f(-m)T(m)$, then there is an $\alpha\in H_{d_1}\cdot H_{d_2}$ such that

$$(d_1d_2)^{j/2}G_{j+1,f}(z_1,z_2)=\frac{w_{d_1}w_{d_2}}{4}\log|\alpha|.$$

This conjecture predicts that Green's-function values at $CM$ points are, up to an explicit algebraic factor, logarithms of algebraic numbers; the source gives no resolution status.