The conjecture that finite Poincaré complexes are manifolds up to homotopy

A finite Poincaré complex is a finite complex satisfying Poincaré duality. Aspherical Poincaré complexes conjecture. Every finite Poincaré complex is homotopy equivalent to a closed manifold. The claim asks whether Poincaré duality alone, together with finiteness, is sufficient for homotopy realization by a closed manifold.