Hausdorff dimension of Besicovitch sets of Cantor graphs

We consider the Hausdorff dimension of planar Besicovitch sets for rectifiable sets $\mathrm{\Gamma }$, i.e. sets that contain a rotated copy of $\mathrm{\Gamma }$ in each direction. We show that for a large class of Cantor sets $C$ and Cantor-graphs $\mathrm{\Gamma }$ built on $C$, the Hausdorff dimension of any $\mathrm{\Gamma }$-Besicovitch set must be at least $\min (2-s^2,\frac{1}{s})$, where $s=\dim C$.