For a field $k$ which satisfies Beilinson-Soule vanishing conjecture, the from Levine's paper,

https://www.uni-due.de/~bm0032/publ/TateMotives.pdf

There exists an abelian category of mixed Tate motives $\text{TM}_k$. If $\sigma:k\rightarrow \mathbb{C}$ is an embedding, we have a Hodge realisation functor, \begin{equation} R_{\sigma}:\text{TM}_k \rightarrow \text{TH}_{\mathbb{Q}} \end{equation} where $\text{TH}_{\mathbb{Q}}$ is the abelian subcategory of $\mathbb{Q}$-mixed Hodge structures generated by Tate objects $\mathbb{Q}$ and is closed under extensions. From Hodge conjecture and conservative conjecture, $R_{\sigma}$ is exact and fully faithful. But since $\text{TM}_k$ is so nice and simple (compared to the category of pure motives or even the conjectured category of mixed motives ), I am wondering whether it has been proved that $R_{\sigma}$ is exact and fully faithful. Any references?