The Hodge-theoretic Gross–Zagier conjecture for Hecke characters

Let $F$ be a quadratic imaginary field, let $\phi$ be a Hecke character of odd weight $w$, and let $H_{\phi}$ be the pure Hodge structure associated with the motive attached to $\phi$. Let $\operatorname{MHS}_{\overline{\mathbb{Q}}}$ be the category of mixed $\mathbb{Q}$-Hodge structures with coefficients in $\overline{\mathbb{Q}}$. Hodge-theoretic Gross–Zagier conjecture. If the sign of the functional equation of $L(\phi,s)$ is $-1$ and $L^\prime(\phi,\frac{w+1}{2})\neq 0$, then

$$\dim_{\mathbb{Q}}\operatorname{Ext}^1_{\operatorname{MHS}_{\overline{\mathbb{Q}}}}\left(\mathds{1},H_{\phi}\left(\frac{w+1}{2}\right)\right)\geq 1.$$

This is presented as the expected Hodge-theoretic analogue of the Gross–Zagier consequence for elliptic curves, predicting a nontrivial extension when the functional-equation sign is negative and the central derivative does not vanish. The supplied passage does not state that it has been proved.