Lang's general Flat Chain Conjecture via the comparison map

For $d\in\mathbb{N}$ and $k\in\mathbb{N}\cup\{0\}$, let $\mathcal{D}_k(\boldsymbol{R}^d)$ denote the metric $k$-currents and let $\overline{\mathcal{D}}_k(\boldsymbol{R}^d)$ denote the corresponding classical currents. The comparison map is the injective linear map

$$C_k\colon\mathcal{D}_k(\boldsymbol{R}^d)\to\overline{\mathcal{D}}_k(\boldsymbol{R}^d).$$

For a compact set $K\subset\boldsymbol{R}^d$, write $\mathbb{F}_k(K)$ for the flat $k$-chains supported in $K$.

Lang's general Flat Chain Conjecture. For every $d\in\mathbb{N}$ and $k\in\mathbb{N}\cup\{0\}$, if $T\in\mathcal{D}_k(\boldsymbol{R}^d)$ has compact support, then

$$C_k(T)\in\mathbb{F}_k(K)$$

for some compact $K\subset\boldsymbol{R}^d$.

This is the general comparison-map formulation of Lang's conjecture. The paper's main result refutes it in the stated nontrivial dimensions, while the remaining cases hold.