Goncharov's motivic higher Chow group conjecture

Let $F$ be a field. For $m\ge 3$, let ${\mathcal G}_m(F)$ be Goncharov's motivic complex group in degree one, let ${\delta}_m$ be its differential, and define

$${\mathcal B}_m(F)=\ker\bigl({\delta}_m:{\mathcal G}_m(F)\to {\mathcal G}_{m-1}(F)\otimes F^\times\bigr).$$

Write $G_{\mathbb Q}=G\otimes\mathbb Q$ for an abelian group $G$. Goncharov's motivic higher Chow group conjecture. For $m\ge 3$,

$${\mathcal B}_m(F)_{\mathbb Q}\cong CH^m(F,2m-1)_{\mathbb Q}.$$

This is proposed as a higher-weight analogue of the known relationship between the Bloch group and $CH^2(F,3)$, but the paper notes that even the case $m=3$ requires modifications and that the relevant relation groups are not yet well understood.