Since there are various models of $\infty$-categories floating around, there are as many models of the associated homotopy 1- and 2-categories. Since the relations between the former are worked out in great detail, it feels like the relations between the latter should be well-understood as well. But since I was unable to find much about the 2-dimensional case I'd like to ask for references here.

Specifically I am interested in the following questions.

1. Given a Kan-enriched category $\mathcal{C}$ we can apply the homotopy coherent nerve to get a quasicategory $N^\Delta(\cal{C})$. We can then apply [HTT Prop. 2.3.4.12] to pass to a simplicial set $h_2(N^\Delta(\cal{C}))$, which Lurie calls the underlying 2-category. On the other hand we can take $\cal{C}$ and apply hom-wise the homotopy-category functor and obtain a Grpd-enriched category, say $H_2(\cal{C})$. We can now apply the Duskin-nerve to get another simplicial set $N^D(H_2(\cal{C}))$, which ought to be a 2-category in the sense of HTT. Are both sSets equivalent in an appropriate sense (I guess it means Joyal-equivalent)?

Even more important to me is the following variation:

2. Note that $H_2(\cal{C})$ is a Grpd-enriched hence a strict (2,1)-category. In the light of Kerodon Rem. 2.3.6 it seems like the Duskin-nerve has a left adjoint $|-|^D:\mathsf{sSet} \rightarrow (2,1)\mathsf{Cat}_{str}$. So are the strict (2,1)-categories $H_2(\cal{C})$ and $|h_2(N^\Delta(\cal{C}))|^D$ equivalent as strict (2,1)-categories or at least as weak (2,1)-categories?

Out of curiosity I'd like to add a third variation on the question:

3. Using the Cordier-realization $|-|^C:\mathsf{sSet} \rightarrow \mathsf{sSet}\text{-}\mathsf{Cat}$ and the fibrant replacement of simplicial categories $R:\mathsf{sSet}\text{-}\mathsf{Cat} \rightarrow \mathsf{Kan}\text{-}\mathsf{Cat}$ we can turn a quasicategory $C$ into a Kan-enriched category $R(|C|^C)$. So how are $|h_2(C)|^D$ and $H_2(R(|C|^C))$ related?

Thank you very much for your time and considerations!

There are various models of $\infty$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in great detail, it feels like the relations between the latter should be well-understood as well. But since I was unable to find much about the 2-dimensional case I'd like to ask for references here.

Specifically I am interested in the following questions.

1. Given a Kan-enriched category $\mathcal{C}$ we can apply the homotopy coherent nerve to get a quasicategory $N^\Delta(\cal{C})$. We can then apply [HTT Prop. 2.3.4.12] to pass to a simplicial set $h_2(N^\Delta(\cal{C}))$, which Lurie calls the underlying 2-category. On the other hand we can take $\cal{C}$ and apply hom-wise the homotopy-category functor and obtain a Grpd-enriched category, say $H_2(\cal{C})$. We can now apply the Duskin-nerve to get another simplicial set $N^D(H_2(\cal{C}))$, which ought to be a 2-category in the sense of HTT. Are both sSets equivalent in an appropriate sense (I guess it means Joyal-equivalent)?

Even more important to me is the following variation:

2. Note that $H_2(\cal{C})$ is a Grpd-enriched hence a strict (2,1)-category. In the light of Kerodon Rem. 2.3.6 it seems like the Duskin-nerve has a left adjoint $|-|^D:\mathsf{sSet} \rightarrow (2,1)\mathsf{Cat}_{str}$. So are the strict (2,1)-categories $H_2(\cal{C})$ and $|h_2(N^\Delta(\cal{C}))|^D$ equivalent as strict (2,1)-categories or at least as weak (2,1)-categories?

Out of curiosity I'd like to add a third variation on the question:

3. Using the Cordier-realization $|-|^C:\mathsf{sSet} \rightarrow \mathsf{sSet}\text{-}\mathsf{Cat}$ and the fibrant replacement of simplicial categories $R:\mathsf{sSet}\text{-}\mathsf{Cat} \rightarrow \mathsf{Kan}\text{-}\mathsf{Cat}$ we can turn a quasicategory $C$ into a Kan-enriched category $R(|C|^C)$. So how are $|h_2(C)|^D$ and $H_2(R(|C|^C))$ related?

Thank you very much for your time and considerations!