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Let $\Omega\subset\mathbb R^{n+1}$ be open and let $E\subset \partial\Omega$with $0<H^s(E)<\infty$, for some $s\in(n,n+1)$, satisfy a local capacitydensity condition. In this paper it is shown that the harmonic measure cannotbe mutually absolutely continuous with $H^s$ on $E$. This answers a question ofAzzam and Mourgoglou, who had proved the same result under the additionalassumption that $\Omega$ is a uniform domain.

The mutual singularity of harmonic measure and Hausdorff measure of codimension smaller than one