In Hanamura's paper Mixed Motives and Algebraic Cycles III

He proved that if assume Grothendieck's standard conjecture, Murre's conjecture and vanishing conjecture, there is a $t$ structure on Voevodsky's category of mixed motives $\textbf{DM}_{\text{gm}}(k,\mathbb{Q})$, whose heart is an abelian category $\text{MM}(k)$. Each of Betti, etale or de Rham cohomology (realisation) functor \begin{equation} \mathbb{R}\Gamma:\textbf{DM}_{\text{gm}}(k,\mathbb{Q}) \rightarrow D^b(\Lambda-\text{Vec}) \end{equation} (for suitable $\Lambda$) is a $t$-exact functor with respect to this $t$ structure and the natural $t$ structure on $D^b(\Lambda-\text{Vec})$. I have several questions,

1, Is the vanishing conjecture Hanamura assumed in his paper stronger or equivalent to the Beilinson-Soule vanishing conjecture about field $k$?

2, Is $\textbf{DM}_{\text{gm}}(k,\mathbb{Q})$ the bounded derived category of $\text{MM}(k)$?, i.e. $\textbf{DM}_{\text{gm}}(k,\mathbb{Q})=D^b(\text{MM}(k))$? Has this been proved in Hanamura's paper or is it still only a conjecture (even with several conjectures assumed)?

3, Let the restriction of $\mathbb{R}\Gamma$ to $\text{MM}(k)$ be \begin{equation} \Gamma:\text{MM}(k) \rightarrow \Lambda-\text{Vec} \end{equation} Since $\mathbb{R}\Gamma$ is $t$ exact, then $\Gamma$ is an exact functor, therefore the derived functor of $\Gamma$ is well defined. \begin{equation} D\Gamma:D^b(\text{MM}(k)) \rightarrow D^b(\Lambda-\text{Vec}) \end{equation} If 2 is true, if $D\Gamma$ the same as (or equivalent to $\mathbb{R}\Gamma$)? Any references or comments will be sincerely appreciated!