Reciprocity law I for pro-semisimple fundamental groups

Assume that $\dim X=1$. Let $P$ be the weight group of $\Delta_{\hat\Pi}$, let $Q\subset P$ be the subgroup generated by the roots, and let $\bar u_\ell\in\operatorname{Hom}((P/Q)^{\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})},\mathbb{Q}/\mathbb{Z})$ and $\bar u_\infty\in\operatorname{Hom}((P/Q)^{\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})},\mathbb{Q}/\mathbb{Z})$ be the classes defined in the text. The sum $\sum_\ell\bar u_\ell$ is well-defined because $(\bar u_\ell,\omega)=0$ for almost all $\ell$ for every $\omega\in(P/Q)^{\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$. Reciprocity law I. One has

$$\sum\limits_\ell\bar u_\ell=\bar u_\infty.$$

This is presented as the first reciprocity law for curves and relates the local classes arising from the pro-semisimple fundamental group to the archimedean contribution; the supplied text gives no resolution status.