Complete metrizability of metrics on the uncountable discrete space

Recall that $\aleph_{1}$ denotes the first uncountable cardinal, and let $D_{\aleph_{1}}$ be the discrete space of cardinality $\aleph_{1}$. The space $\mathrm{Met}(D_{\aleph_{1}})$ consists of metrics on this space, equipped with the topology considered in the paper. The uncountable discrete-space conjecture. The space $\mathrm{Met}(D_{\aleph_{1}})$ is not completely metrizable. This conjecture is proposed to remove the separability assumption from the known characterization of complete metrizability for $\mathrm{Met}(X)$; the separable case is known to be equivalent to $X$ being $\sigma$-compact, while the nonseparable case remains open.