Singer's $L^2$-Betti-number conjecture

Let $M^n$ be a closed aspherical manifold. Singer's conjecture. The rational $L^2$-Betti numbers of its fundamental group vanish outside the middle dimension:

$$b_i^{(2)}(\pi_1(M^n);\mathbb{Q})=0\quad\text{for }i\ne\frac n2.$$

This is a basic vanishing principle for computing $L^2$-Betti numbers and would force the free part of homology to grow sublinearly outside the middle dimension in residually finite settings. The source attributes the conjecture to Singer but gives no resolution status.