Orponen's direction-set conjecture for planar Borel sets

Let $E$ be a Borel set in the plane that is not contained in a line, and define its direction set by

S(E)=\left\\{\frac{x-y}{|x-y|}:x,y\in E,\ x\ne y\right\\}.

Orponen's conjecture.

$$\dim_{\mathcal{H}}S(E)=\min\\{\dim_{\mathcal{H}}E,1\\}.$$

This conjecture predicts the maximal possible Hausdorff dimension of the set of directions determined by a planar set. The paper describes its theorem as an $\epsilon_0$-increment toward the conjecture, so the claim remains open there.