Cheeger's measurable non-degeneracy conjecture for coordinate charts

Let $(X,\rho,\mu)$, $\{X_n\}_{n=1}^\infty$ and $\xi^n: X \to \mathbb{R}^{k(n)}$ be as in the paper's differentiability theorem. Here $X^n$ is the corresponding coordinate-chart domain, $\xi^n(X^n)$ is its image, and $\mathcal{H}^{k(n)}$ denotes $k(n)$-dimensional Hausdorff measure.

Cheeger's conjecture.

$$\mathcal{H}^{k(n)}(\xi^n(X^n)) > 0.$$

The conjecture asserts that coordinate-chart images are measurably non-degenerate. If true, it would allow Cheeger's rectifiability theorem to apply globally to these images; the supplied text gives no resolution status.