The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure

Let $\mathcal{N}^h_Ω$ be the null ideal of the Hausdorff measure constructed by Davies and Rogers. We give a Tukey reduction of $(\mathcal{N}^h_Ω,\mathcal{N}^h_Ω,\subseteq)$ to a localization system with finite coordinate sets. It follows that $\mathfrak{v}^\forall_{D,g}\le\operatorname{add}(\mathca