I have frequently come across the statement "Any two triangulations of a compact n-manifold are related by bistellar moves" attributed to Pachner via Lickorish's paper 'Simplicial moves on complexes and manifolds'. The theorem this refer to is the following: Closed combinatorial n-manifolds are PL homeomorphic if and only if they are bistellar equivalent.

My question is: Considering that Hauptvermutung is not true for manifolds of dimension more than 3, how can we justify this statement?