On the Hausdorff measure of projections of self-similar sets

For $0<d\leq1/4$, let $\mathcal{C}(d)$ be the four-corner Cantor set with Hausdorff dimension $s_d=\log 4/\log(1/d)$. Peres, Simon, and Solomyak, as well as Mattila, asked for which $d$ the measure $\Hau^{s_d}(p_θ(\mathcal{C}(d)))$ is positive for almost every direction $θ$. It was open for the rang