Colmez's original height formula conjecture

Let ${\mathcal{CM}}^0$ be the space of class functions associated with CM-types, and let $ht$ and $Z(\mathord\cdot,s)$ be the linear height and logarithmic-derivative functionals defined on it. For $a\in{\mathcal{CM}}^0$, define its dual by

$$a^\vee(g)=a(g^{-1})$$

for $g\in\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$. Colmez's original conjecture. For every $a\in{\mathcal{CM}}^0$,

$$ht(a)=-Z(a^\vee,0).$$

This is Colmez's functional formulation of the relationship between CM Faltings heights and logarithmic derivatives of Artin $L$-functions. The source gives no resolution status for the full conjecture; related averaged versions are known, but the asserted identity for every class function remains the target in the cited discussion.