Kielak and Fisher have connected the $L^2$-Betti numbers (and their finite field variants) to the Novikov homology for a RFRS group $G$. This in turn relates vanishing of $F$-$L^2$-Betti numbers of $G$ to algebraic virtual fibering of $G$. We will give an example of a RFRS group $G$ which has vanishing top-dimensional Novikov cohomology with all field coefficients but not with $\mathbb{Z}$-coefficients.