Wall's realization conjecture for Poincaré duality groups

Let $\Gamma$ be a finitely presented $n$-dimensional Poincaré duality group over $\mathbb Z$. Wall's conjecture. There exists a closed $n$-dimensional manifold $M$ which is $K(\Gamma,1)$. This problem is open for all $n\ge 3$; the cases $n=1$ and $n=2$ are known.