The Singer conjecture for Poincaré duality groups

Let $\Gamma$ be an $n$-dimensional Poincaré duality group, and let $b_i^{(2)}(\Gamma)$ denote its $i$th $\ell^2$-Betti number. Singer conjecture. The $\ell^2$-Betti numbers vanish away from the middle degree:

$$b_i^{(2)}(\Gamma)=0\qquad\text{for }i\neq n/2.$$

This is presented as the best-known conjecture concerning $\ell^2$-homology. The source gives no resolution, so the conjecture is open.