Further to my question,

A Naive Question on Mixed Motives and Mixed Hodge Structures

that has received very good replies and suggestions, and I really appreciate it. I am going to ask a question on generalised Hodge conjecture which is closely related to last one. From conjecture 3.22 of Marc Levine's Mixed Motives in K-theory hand book

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

it conjectures a functor from Nori's mixed motives to rational MHS, \begin{equation} H:\mathcal{MM}_{\text{Nori}}(k,\mathbb{Q})\rightarrow \text{MHS}_{\mathbb{Q}} \end{equation} is fully faithful, which could be seen as the generalisation of Hodge conjecture. My first question is why it is Nori's mixed motives that fits into this conjecture?

If $H$ is conjectured to be fully-faithful, then (I guess) the derived functor of $H$ (by abuse of notation) \begin{equation} DH:D^b(\mathcal{MM}_{\text{Nori}}(k,\mathbb{Q}))\rightarrow D^b(\text{MHS}_{\mathbb{Q}}) \end{equation} is also fully faithful. In D. Harrer's PhD thesis, Comparison of the Categories of Motives defined by Voevodsky and Nori

https://arxiv.org/ftp/arxiv/papers/1609/1609.05516.pdf

From main theorem, 7.4.17, there exists a realization functor \begin{equation} R_{\text{Nori}}: DM_{gm}(k,\mathbb{Q})\rightarrow D^b(\mathcal{MM}_{\text{Nori}}(k,\mathbb{Q})) \end{equation} I guess the composition of $R_{\text{Nori}}$ and $DH$ \begin{equation} DH \circ R_{\text{Nori}}:DM_{gm}(k,\mathbb{Q}) \rightarrow D^b(\text{MHS}_{\mathbb{Q}}) \end{equation} is the usual Hodge realisation functor of Voevodsky's motive. My second question is, is there a generalised Hodge conjecture stated using Voevodsky's category instead of Nori's mixed motive? e.g. like $DH \circ R_{\text{Nori}}$ is fully faithful?

Any references, comments and answers will be fully faithfully appreciated!