Disjoint subsets $\mathbb{R}^n$ with same Hausdorff measure on open sets

Question

Can we choose sets $X_1,X_2 \subset \mathbb{R}^n$ be two disjoint sets with Hausdorff dimension $0\le d \le n$, such that $0<\mathcal{H}^d(X_1)=\mathcal{H}^d(X_2)<\infty$, such that for any open set $U \subset \mathbb{R}^n$, we have $\mathcal{H}^d(U \cap X_1)=\mathcal{H}^d(U \cap X_2)$? I know this is false for Lebesgue measure, but I don't know if a similar density restriction applies to Hausdorff measure.