Exceptional-set estimate for visible parts

Let $K \subset \mathbb{R}^{2}$ be compact, let $\alpha \in[0,1]$, and let $\operatorname{vis}_{\sigma}(K)$ be the visible part of $K$ in direction $\sigma$. Exceptional-set estimate for visible parts.

$$\mathcal{H}^{1}(\operatorname{vis}_{\sigma}(K)) \leq 2-\alpha$$

for $\mathcal{H}^{\alpha}$-almost every $\sigma \in[-1,1]$. This is a stronger parameterized prediction controlling the exceptional directions according to their Hausdorff measure dimension; the source provides no resolution, so it remains open.