Invariance of Hausdorff null-set cardinal invariants for compact Polish spaces

Let $X$ be a compact Polish space, let $\mathcal{H}^s$ denote the $s$-dimensional Hausdorff measure, and let $\mathrm{HDZ}_X$ be the corresponding ideal of Hausdorff-dimension-zero subsets of $X$. Write $\mathrm{HDZ}$ for the associated ideal on the standard reference space, and for an ideal $\mathcal{I}$ let $\operatorname{non}(\mathcal{I})$ and $\operatorname{cov}(\mathcal{I})$ denote its uniformity and covering cardinal invariants.

Invariance conjecture. For every compact Polish space $X$ with

$$0 < \mathcal{H}^s(X) < \infty$$

for some $s>0$,

$$\operatorname{non}(\mathrm{HDZ}_X)=\operatorname{non}(\mathrm{HDZ})$$

and

$$\operatorname{cov}(\mathrm{HDZ}_X)=\operatorname{cov}(\mathrm{HDZ}).$$

The claim proposes that these cardinal characteristics do not depend on the particular compact Polish space once it carries a finite, positive Hausdorff measure in some positive dimension. The supplied text establishes the analogous equalities for Euclidean spaces, but gives no resolution of this broader compact-Polish-space assertion.