The flat chain conjecture for metric currents

Let $k\in\mathbb{N}$, let $\mathrm{X}$ be a metric space, and let $\mathbf{M}_k(\mathrm{X})$ denote the space of metric $k$-currents on $\mathrm{X}$. Normal $k$-currents are metric $k$-currents whose mass and boundary mass are finite, and the flat norm is the metric induced by flat decompositions. Flat chain conjecture. Every $k$-current on a metric space $\mathrm{X}$ can be approximated by normal $k$-currents under the flat norm, namely

$$\overline{\{\text{normal }k\text{-currents}\}}^{\text{Flat norm}}=\mathbf{M}_k(\mathrm{X}).$$

This conjecture asks whether metric currents are flat limits of normal currents. It is known in Euclidean space for $k=1$ and $k=n$, and Schioppa proved the corresponding metric statement for normal $1$-currents in quasiconvex spaces satisfying a finite-dimensionality assumption; the general metric version remains open.