Bloch group conjecture for $CH^3(F,5)$

Let $F$ be a field. Let $B_2(F)$ be the Bloch group, let $R_3(F)$ be the subgroup generated by the stated relations in ${\mathbb Z}[{\mathbb P}^1_F]$, and define

$$B_3(F)=\ker\bigl(\beta_3:{\mathbb Z}[{\mathbb P}^1_F]\to B_2(F)\otimes F^\times\bigr)/R_3(F).$$

Let ${\mathcal CH}^3(F,5)_{\mathbb Q}$ denote the modified higher Chow group introduced in the paper. Bloch group conjecture for $CH^3(F,5)$. One has

$$B_3(F)_{\mathbb Q}\cong {\mathcal CH}^3(F,5)_{\mathbb Q}\cong CH^3(F,5)_{\mathbb Q}.$$

The statement is presented as the ultimate goal of the work. The paper notes that the first isomorphism involves the modified group and that identifying it with the ordinary higher Chow group depends on a further mild conjecture.