Structure of 2-skeletons of higher dimensional regular polytopes

In our paper [2], we have classified simple regular polyhedral BP-complexes, which are polyhedral complexes satisfying certain natural conditions on their vertex structures. As an addendum of this classification, we prove that 2-skeletons of higher dimensional regular polytopes are simple regular polyhedral complexes, but not polyhedral BP-complexes. The proof is done by a detailed investigation of their vertex structures.