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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$ be open and let $E\subset \mathrm{\partial }\mathrm{\Omega }$ with $0<H^s(E)<\mathrm{\infty }$, for some $s\in (n,n+1)$, satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with $H^s$ on $E$. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that $\mathrm{\Omega }$ is a uniform domain.