Does anyone know of a simplicial complex which is not collapsible but whose barycentric subdivision is?

Every collapsible complex is necessarily contractible, and subdivision preserves the topological structure, so we are certainly looking for a complex which is contractible, but not collapsible. The only complexes I know of which are contractible but not collapsible are the dunce cap and Bing's house with two rooms. Neither of these have any free faces, and so no iterated subdivision will result in a collapsible complex.