I am reading the book Periods and Nori Motives by Huber and Muller-Stach et al. A question comes up to me.
Suppose $\text{DM}_{gm}(k,\mathbb{Q})$ is the triangulated category of geometric mixed motives constructed by Voevodsky, assume now the field $k$ satisfies the vanishing conjecture of Soule and Beilinson. Denote $\text{DTM}_k$ the triangulated subcategory generated by Tate objects $\mathbb{Q}(n),\,n \in \mathbb{Z}$, then there is a $t$ structure on $\text{DTM}_k$ whose heart is the abelian category of Tate motives, denoted by $\text{TM}_k$.
In the category of Nori's mixed motive, $\text{NN}_{Nori}(k,\mathbb{Q})$, and the Tate objects $\mathbf{1}(n), n\in \mathbb{Z}$ in this abelian category generate the mixed Nori Tate motives $\text{TM}_{Nori,k}$, whose derived category will be denoted by $\text{DTM}_{Nori,k}$. From this book there is a fully faithful functor \begin{equation} H:\text{TM}_k \rightarrow \text{TM}_{Nori, k} \end{equation}
My question is, when is $H$ an equivalence of the two abelian categories? We know when $k$ is a number field, the vanishing conjecture is satisfied, in this case is $H$ an equivalence of categories? References will be fully faithfully appreciated.
Edit: for $X$ and $Y$ in $\text{TM}_k$, do we have \begin{equation} \text{Ext}^1_{\text{TM}_k}(X,Y)=\text{Ext}^1_{\text{TM}_{Nori,k}} ~~(X,Y) \end{equation} ?
