The visibility conjecture for visible parts of compact sets

Let $E\subset\mathbb{R}^d$ be compact. For $\theta\in\mathbb{S}^{d-1}$, let

$$\ell_\theta=\{t\theta:t\ge 0\}$$

be the closed half-line spanned by $\theta$, and define the visible part of $E$ in direction $\theta$ by

$$\operatorname{Vis}_\theta(E)=\{x\in E:(x+\ell_\theta)\cap E=\{x\}\}.$$

Write $\operatorname{dim}_{\mathrm{H}}$ for Hausdorff dimension. Visibility conjecture. If $\operatorname{dim}_{\mathrm{H}}(E)>d-1$, then for $\mathcal{H}^{d-1}$-almost every $\theta\in\mathbb{S}^{d-1}$,

$$\operatorname{dim}_{\mathrm{H}}(\operatorname{Vis}_\theta(E))=d-1.$$

For sets with Hausdorff dimension at most $d-1$, the corresponding equality follows from the projection theorem and the trivial upper bound. The conjecture concerns the remaining case; the paper proves upper bounds strictly below the dimension of $E$ for almost every direction, but does not establish the conjectured value in general.