The question is the title.

In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are $2$-functors. If $\mathcal{A}$ is a $2$-category, then $\pi_{0}(\mathcal{A})$ is the category whose objects are those of $\mathcal{A}$ and whose $1$-cells are obtained from those of $\mathcal{A}$ by identifying two $1$-cells if they are linked by a zigzag of $2$-cells in $\mathcal{A}$. (In other words, the $Hom$ sets of $\pi_{0}(\mathcal{A})$ are given by the $\pi_{0}$ of the $\underline{Hom}$ categories of $\mathcal{A}$.)

Dwyer-Kan equivalences were defined by Dwyer and Kan in the general setting of simplicial categories. In the realm of $2$-categories, a map $u : \mathcal{A} \to \mathcal{B}$ is a Dwyer-Kan equivalence if the two following conditions are satisfied:

$(i)$ For every objects $X$ and $Y$ of $\mathcal{A}$, the functor $\underline{Hom}_{\mathcal{A}}(X,Y) \to \underline{Hom}_{\mathcal{B}}(u(X),u(Y))$, induced by $u$, `is a weak equivalence (which means that its nerve is a simplicial weak equivalence or, which is equivalent, that this induced functor belongs to any basic localizer of $Cat$).

$(ii)$ The functor $\pi_{0}(\mathcal{A}) \to \pi_{0}(\mathcal{B})$, induced by $u$, is essentially surjective.

Note that these conditions imply that not only is $\pi_{0}(u)$ essentially surjective, but it is also an equivalence of categories.

Some people have apparently suggested that the localization of $2-Cat$ with respect to the class of Dwyer-Kan equivalences should give, up to equivalence, the category of $(\infty,1)$-categories. However, I have yet to find someone who could point out a proof in the literature or write a proof themselves when asked the question whether this result is more than folkloric belief. Could somebody provide something more concrete? Note that I have no definite clue whether this result is true or not. Arguments against its validity would be welcome without dismay.

2more comments