Gross–Zagier and Gross–Kohnen–Zagier conjecture for CM values of weight-4 automorphic Green's functions

Let $N\in\\{1,2,3,4\\}$, and let $z,z'\in\mathfrak H$ be quadratic irrational points satisfying

$$[\mathbb Q(z):\mathbb Q]=[\mathbb Q(z'):\mathbb Q]=2.$$

Assume that $z\notin\varGamma_0(N)z'$, where $\varGamma_0(N)$ is the Hecke congruence group of level $N$, and let $G^{\mathfrak H/\overline{\varGamma}_0(N)}_{2}(z,z')$ denote the weight-4 automorphic Green's function. Gross–Zagier and Gross–Kohnen–Zagier conjecture. The CM values are always expressible as logarithms of algebraic numbers, namely

$$\exp\left[\operatorname{Im} z\operatorname{Im} z'G^{\mathfrak H/\overline{\varGamma}_0(N)}_{2}(z,z')\right]\in\overline{\mathbb Q}.$$

This conjecture predicts an algebraicity phenomenon for automorphic Green's functions evaluated at pairs of CM points at the four levels for which the weight-4 cusp-form-free condition holds. The source presents it as an outstanding problem; no resolution is supplied here.