Consider a category $C$ enriched in categories. Here we have a natural class $W$ of equivalences, namely those morphisms having inverses up to 2-isomorphisms. One can take the hammock localization $L^H(C,W)$ and get a simplicial category.

On the other hand one can take the maximal subgroupoid of each $hom$-category (or simply suppose that the hom-categories were groupoids from the start, this wouldn't change the above notion of equivalence), apply the nerve functor $hom$-wise and get another simplicial category.

Are these two simplicial categories Dwyer-Kan equivalent?

[I am also interested in a particular case of this question, which would take too long to expose here, which has the feature that the $hom$-categories are preorders, and hence the maximal subgroupoids are simply equivalence relations on the sets of morphisms (i.e. the sets of objects of the hom-categories). Yet I couldn't see by hand that the hom-simplicial sets of the hammock localization are homotopy discrete...]