Proximally well-monotone covers and QH-singularity

We use a certain class of well-monotone covers on a quasi-uniform space \({(X, \mathcal{U})}\) to investigate whether there are quasi-uniformities \({\mathcal{V}}\) that are distinct from \({\mathcal{U}}\), but have the property that the associated Hausdorff quasi-uniformities \({\mathcal{U}_H}\) and \({\mathcal{V}_H}\) on the hyperspace of X have the same underlying topologies.