Okun-Schreve conjecture on L2-Betti numbers of aspherical manifolds with boundary

Let $(M,\partial M)$ be a compact aspherical $n$-manifold-with-boundary, and write $b^{(2)}_{>n/2}(M)=0$ to mean that all $L^2$-Betti numbers in degrees greater than $n/2$ vanish. Okun-Schreve's conjecture.

$$b^{(2)}_{>n/2}(M)=0.$$

This is the restricted version of the Davis-Okun vanishing conjecture discussed in the source and is related there to the Singer conjecture; the source does not state its resolution.