  Given a Coxeter system $(W,S)$ there is a contractible simplicial complex
$\Sigma$ called the Davis complex on which $W$ acts properly and cocompactly.
In an article of Dymara, the weighted $L^2$-(co)homology groups of $\Sigma$
were defined, and in an article of Davis-Dymara-Januszkiewicz-Okun, the Singer
conjecture for Coxeter groups was appropriately formulated for weighted
$L^2$-(co)homology theory. In this article, we prove the weighted version of
the Singer conjecture in dimension three under the assumption that the nerve of
the Coxeter group is not dual to a hyperbolic simplex, and in dimension four
under additional restrictions. We then prove a general version of the
conjecture where the nerve of the Coxeter group is assumed to be a flag
triangulation of a $3$-manifold.
