The generalized Singer conjecture for weighted $L^2$-homology

Let $(W,S)$ be a Coxeter system with nerve $L$, let $Sigma$ be its associated complex, and suppose that $L$ is a generalized homology sphere of dimension $n-1$, written $GHS^{n-1}$. Let ${\mathbf q}$ be a positive weight vector, with ${\mathbf q}\leq{\mathbf 1}$ coordinatewise. The generalized Singer conjecture. If $k>\frac{n}{2}$, then

$$L^2_{\mathbf q}\mathcal H_k(\Sigma)=0.$$

By Poincare duality, the source notes the equivalent formulation for ${\mathbf q}\geq{\mathbf 1}$ and $k<\frac{n}{2}$. This extends the ordinary Singer conjecture to weighted $L^2$-homology; evidence is given in several low-dimensional and right-angled cases, while the general assertion remains open.