Zero-in-the-spectrum conjecture for finite aspherical complexes

Let $M$ be a finite aspherical $CW$-complex (or weakly a closed, connected, oriented and aspherical Riemannian manifold) with fundamental group $\pi$. Write $C_{r}^{\ast }(\pi )$ for the reduced group $C^{\ast }$-algebra of $\pi$, and let $X$ denote the space used in the paper's notation. Zero-in-the-spectrum conjecture. There is some $i\geq 0$ such that

$$H_{i}(X;C_{r}^{\ast }(\pi ))\neq 0.$$

Equivalently, zero belongs to the spectrum of the relevant Laplacian; the conjecture asserts that this homology does not vanish in every degree.