Gross–Kohnen–Zagier algebraicity conjecture for higher Green's functions

Let $k\geq 2$, let $f\in M^!_{2-2k}$ be a weakly holomorphic modular form with rational Fourier coefficients, and let $G_{k,f}$ be the higher Green's function associated to $f$. Write $X_\Gamma=\Gamma\backslash\mathbb{H}$ for the modular curve with $\Gamma=\operatorname{SL}_2(\mathbb{Z})$. For a CM point $Z=(z_1,z_2)\in X_\Gamma^2$ outside the singularity $T_f$ of $G_{k,f}$, let $d_j$ be the discriminant of $z_j$. Gross–Kohnen–Zagier conjecture. There exists $\alpha\in\overline{\mathbb{Q}}\subset\mathbb{C}$ such that

$$G_{k,f}(Z)=-(d_1d_2)^{\tfrac{1-k}{2}}\log|\alpha|.$$

This conjecture predicts the algebraic nature of special values of higher Green's functions at CM points, extending the algebraicity phenomenon for singular moduli. The paper proves an averaged version and studies the factorization of the ideal generated by these algebraic values, while the stated pointwise assertion is not resolved here.