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I assume that your measure is defined as $$ \mathcal H^s\otimes\mathcal H^r(S) := \inf\left\{\sum_{k=0}^\infty\mathcal H^s(A_k)\mathcal H^r(B_k)\middle|A\in\mathcal B(\mathbb R^n)^{\mathbb N},B\in\...

Yes, this is true. It holds even under the weaker assumptions that $X$ is compact and connected with finite one-dimensional Hausdorff measure. Alberti and Ottolini (On the structure of continua with ...