Algebraicity conjecture for higher Green functions at CM points

Let $Z(W_\mathbb{Q})$ be the CM cycle and let $\Phi^r_L(z,h,f)$ be the higher Green function in the setting of Theorem. Assume that $f$ has integral Fourier coefficients and that the singularity of $\Phi^r_L(z,h,f)$ does not intersect $Z(W_\mathbb{Q})$. Then there are $\lambda_j\in F$ and $\alpha_j\in H$ for $1\leq j\leq d$ such that

$$\Phi^r_L(z_0,h,f)=\sum_{j=1}^d\lambda_j\log\left|\sigma_h(\alpha_j)\right|$$

for every $(z_0,h)\in Z(W_\mathbb{Q})$. This conjecture predicts an algebraic logarithmic description of higher Green-function values at CM points, extending the preceding results beyond the cases $n\leq 2$ or $d=1$; the stated setting includes cases with $n\geq 3$ and $d\geq 2$, for which no corresponding result is known even after averaging over CM points and their Galois conjugates.