The visibility conjecture for visible parts of planar compact sets

Let $K \subset \mathbb{R}^{2}$ be compact. For $\sigma \in[-1,1]$, let $\operatorname{vis}_{\sigma}(K)$ denote the points of $K$ that are not followed by another point of $K$ along the ray of slope $\sigma$. Visibility conjecture.

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

for $\mathcal{H}^{1}$-almost every $\sigma \in[-1,1]$. This predicts a universal one-dimensional bound for visible parts in almost every direction; its resolution is not supplied in the source.