This question is related with this one. For simplicial complex (which we have to assume is ordered as explained in the answer of the linked question) we have a construction of geometric realization which is defined as all formal convex combinations of vertices. For such simplicial complex we have the construction of the associated simplicial set (see the linked question) and for such simplicial set there is also the construction of geometric realization. In the linked question it was asked whether the homology of simplicial complex is isomorphic with the homology of the associated simplicial set. One can show that both simplicial homologies are isomorphic with the singular homology of geometric realizations. One could hope that it is possible to prove that simplicial homologies are isomorphic by proving that their geometric realizations are homotopy equivalent. So this is my question:
Given an ordered simplicial complex $K$ consider the associated simplicial set $Ss(K)$. Consider geometric realizations: $|K|$ and $|Ss(K)|$. Is it true that $|K|$ and $|Ss(K)|$ are homotopy equivalent?