The refined Colmez conjecture for individual CM periods

Let $A$ be a CM abelian variety over a number field $K$, let ${\cal A}$ be its integral model, and let $\omega_{\cal A}$ be the associated Hodge line bundle. Let $E$ be a CM field of degree $2d$, let $K$ contain $E$, and let $G={\rm Gal}(E|{\Bbb Q})$. Let $\Phi$ be the CM type of $A$, let $\Phi^{\vee}(\tau)=\Phi(\tau^{-1})$, and for complex-valued functions on $G$ define

$$\langle f,g\rangle={1\over \#G}\sum_{\gamma\in G}f(\gamma)\overline{g(\gamma)},\qquad (f\ast g)(\lambda)={1\over \#G}\sum_{\gamma\in G}f(\gamma)g(\gamma^{-1}\lambda).$$

For each odd irreducible Artin character $\chi$ of $G$, write $f_{\chi}$ for its conductor and $L(\chi,s)$ for its Artin $L$-function.

Refined Colmez conjecture. There exists an integer $m\geq1$ and an element $\beta=\beta(A,m,\phi)\in\omega_{\cal A}^{\otimes m}$ such that (a) $\beta$ generates $\omega_{\cal A}^{\otimes m}$, in particular $\omega_{\cal A}^{\otimes m}\simeq{\mathcal O}_K$; and (b) for every $\sigma\in{\rm Gal}(K|{\Bbb Q})$ and every nonzero $\alpha\in\omega_{\cal A}$,

$$\left|(2\pi)^{-d}\int_{{\cal A}_{\sigma}({\Bbb C})}\alpha_{\sigma}\wedge\bar{\alpha}_{\sigma}\right|^{1/2}=|\sigma(\alpha^{\otimes m}/\beta)|^{1/m}\exp\left(d\sum_{\chi\ {\rm odd}}\langle\Phi\ast\Phi^{\vee},\chi\rangle\left[2{L'(\chi,0)\over L(\chi,0)}+\log(f_{\chi})\right]\right).$$

This refinement seeks to identify the individual transcendental terms whose average gives the Faltings height, and would imply fine reciprocity laws for Siegel modular forms at CM points. The source proves it for elliptic curves using results on elliptic units; the general assertion is not assigned a resolution status here.