Cantor sets with absolutely continuous harmonic measure

We construct Ahlfors regular Cantor sets $K$ of small dimension in the plane, such that the Hausdorff measure on $K$ is equivalent to the harmonic measure associated to its complement. In particular the Green function in $R^2 \backslash K$ satisfies $G^p (x) \simeq \mathrm{dist} (x, K)^\delta$ whenever $\mathrm{dist} (x, K) \le 1$ and $p$ is far from $K$.