Colmez's product formula for CM abelian varieties

Let $X$ be a CM abelian variety over a number field $K$, let $E$ be its CM algebra with CM-type $\Phi$, let $H_E=\operatorname{Hom}_Q(E,Q^{\rm alg})$, and assume the notation $\omega_\psi^\eta$, $u_\eta$, and $\langle \omega_\psi^\eta,\omega_{c\psi}^\eta,u_\eta\rangle_v$ is as above, with $v$ ranging over the places of $Q$ and $\eta$ over the embeddings of $K$. Colmez's product formula. The sum in the preceding regularized expression should be zero, equivalently

$$\prod\limits_v\prod\limits_{\eta\in H_K}\bigl|\langle \omega_\psi^\eta,\omega_{c\psi}^\eta,u_\eta\rangle_v\bigr|_v=1.$$

This is the product-formula form of Colmez's period conjecture; the supplied text gives no resolution status for this formulation.