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No it is not. But there is a replacement for this. There is another model of $\infty$-categories: marked simplicial sets. If $X$ is a simplicial set with a set of marked $1$-simplices $S$, the fibrant replacement of $(X,S)$ is a model of the localization by $S$ of the $\infty$-category corresponding to $X$ (as a mere simplicial set). There is always a canonical map $\tau_K:\Delta_{/K}\to K$ sending an $n$-simplex over $K$ to its value at $n$. We can mark those $1$-simplices of $\Delta_{/K}$ that are sent to the identites in $K$. Then the map $\tau_K:\Delta_{/K}\to K$ is an equivalence in the model structure of marked simplicial sets (where the marked simplices in $K$ are the identities). By a 2 out of 3 property, a morphism of simplicial sets $K\to K'$ is a categorical equivalence if and only if the induced functor $\Delta_{/K}\to\Delta_{/K'}$ is a weak equivalence of marked simplicial sets. The main properties of the comparison map $\tau_K:\Delta_{/K}\to K$ needed to prove what I clain above are proved in Kerodon or in my book (Prop. 7.3.15) for instance. The model structure on marked simplicial sets is discussed in HTT, of course.