Metric realisation conjecture for first Čech homology

Let $X$ be a Peano continuum, that is, a compact, connected, locally connected metrizable space. For a compatible metric $d$ on $X$, let $\hat{H}_d$ denote the associated metric-dependent homology group. Then metric realisation conjecture. $X$ has a metric compatible with its topology such that the corresponding $\hat{H}_d$ coincides with the first Čech homology group of $X$. This would produce a metric choice for which the paper's metric-dependent homology agrees with a standard topological homology theory. The source presents this as a conjecture motivated by possible higher-dimensional analogues; no resolution is given.