Proximally well-monotone covers and $QH$-singularity

In this paper 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.