Coarse metrizability of proper bounded coarse structures

Let $X$ be a set and let ${\operatorname{D}}$ be a family of pseudometrics on $X$. The topology induced by ${\operatorname{D}}$ is the topology generated by the pseudometric balls, and $\mathcal{E}_b({\operatorname{D}})$ is the bounded coarse structure induced by ${\operatorname{D}}$. Assume that the induced topology is locally compact and that $\mathcal{E}_b({\operatorname{D}})$ is proper. Coarse metrizability conjecture. The coarse space $(X,\mathcal{E}_b({\operatorname{D}}))$ is coarsely metrizable: there exists a pseudometric $\operatorname{d}$ on $X$ such that

$$\mathcal{E}_b({\operatorname{D}})=\mathcal{E}_b({\operatorname{d}}).$$

The claim asks whether local compactness of the topology together with properness of the bounded coarse structure forces the structure to arise from one pseudometric. The paper immediately states that a counterexample shows the conjecture is false.