Williamson–Janos conjecture on complete admissible metrics

A complete admissible metric $d$ for a $\sigma$-compact, locally compact space $X$ is a metric inducing the topology of $X$ and complete as a metric space. A Heine–Borel metric is a metric for which every closed bounded set is compact. The conjecture asserts that, whenever

$$\operatorname{Cl}\{x\in X\mid d(x,x_0)<r\}=\{x\in X\mid d(x,x_0)\leq r\},\qquad \forall x_0\in X,\ \forall r>0,$$

a complete admissible metric $d$ is always a Heine–Borel metric. This topological conjecture would make the quasi-local compactness hypothesis in the paper unnecessary; the source gives no indication that it has been resolved.