Conjecture on the dimension of directions spanned by planar sets

Let $K\subset\mathbb{R}^{2}$ be a Borel set, not contained on a line, and define the set of unit vectors spanned by $K$ by

$$S(K):=\left\{\frac{x-y}{|x-y|}\in S^{1}:x,y\in K,\ x\neq y\right\}.$$

Here $\dim_{\mathrm{H}}$ denotes Hausdorff dimension. Direction-set dimension conjecture.

$$\dim_{\mathrm{H}}S(K)=\min\{\dim_{\mathrm{H}}K,1\}.$$

This would sharpen the preceding lower bound $\dim_{\mathrm{H}}S(K)\geq \dim_{\mathrm{H}}K/2$ and predict the maximal dimension allowed by the fact that $S(K)\subset S^{1}$.