The conjecture 51 in Levine's Mixed Motives in Handbook of K-theory

http://www.math.illinois.edu/K-theory/handbook/1-429-522.pdf

states that the functor induced by $hs:\text{ECM} \rightarrow \text{MHS}$, denoted by, \begin{equation} \mathfrak{hs}:\text{NMM}(k)_{\mathbb{Q}} \rightarrow \text{MHS}_{\mathbb{Q}} \end{equation} where $\text{NMM}(k)_{\mathbb{Q}}$ is Nori's mixed motives over a field $k$ (which admits an embedding $\sigma: k \rightarrow \mathbb{C}$) tensored by $\mathbb{Q}$. I am not clear with the construction of the functor $\mathfrak{hs}$ (I guess this is the Hodge realisation functor of Nori's mixed motives), could anyone give me some references? I also have two questions about the properties of $\mathfrak{hs}$.

Is $\mathfrak{hs}$ (conjectured to be) exact or not? Has the exactness of it been proved?

If $\mathfrak{hs}$ is exact, then it has a derived functor, \begin{equation} D\mathfrak{hs}:D^b(\text{NMM}(k)_{\mathbb{Q}}) \rightarrow D^b(\text{MHS}_{\mathbb{Q}}) \end{equation} In Harrer's thesis,

https://arxiv.org/abs/1609.05516

he constructs a functor,

\begin{equation} R_n:\text{DM}_{gm}(k,\mathbb{Q}) \rightarrow D^b(\text{NMM}(k)_{\mathbb{Q}}) \end{equation} The composition of it with $D\mathfrak{hs}$ is a functor

\begin{equation} D\mathfrak{hs} \circ R_n :\text{DM}_{gm}(k,\mathbb{Q}) \rightarrow D^b(\text{MHS}_{\mathbb{Q}}) \end{equation} Is this functor equivalent to the Hodge realisation functor? Any references?