Dimension of elliptic harmonic measure of Snowspheres

A metric space $\mathcal{S}$ is called a \defn{quasisphere} if there is a quasisymmetric homeomorphism $f\colon S^2\to \mathcal{S}$. We consider the elliptic harmonic measure, i.e., the push forward of 2-dimensional Lebesgue measure by $f$. It is shown that for certain self similar quasispheres $\mathcal{S}$ (snowspheres) the dimension of the elliptic harmonic measure is strictly less than the Hausdorff dimension of $\mathcal{S}$.