On the homotopy hypothesis in dimension 3

We show that if the canonical left semi-model structure on the category of Grothendieck $n$-groupoids exists, then it satisfies the homotopy hypothesis, i.e. the associated $(\infty,1)$-category is equivalent to that of homotopy $n$-types, thus generalizing a result of the first named author. As a corollary of the second named author's proof of the existence of the canonical left semi-model structure for Grothendieck 3-groupoids, we obtain a proof of the homotopy hypothesis for Grothendieck 3-groupoids.