Plane visible part conjecture

Prove that for every compact set E ⊂ ℝ^2 with Hausdorff dimension at least 1, the visible part Vis_θ(E) has Hausdorff dimension 1 for Lebesgue-almost every direction θ.

Background

The visible part Vis_θ(E) consists of points in E that are first encountered by half-lines in direction θ, modeling what is visible from infinity. Marstrand’s projection theorem implies that if dim_H(E) ≥ 1, then dim_H(Vis_θ(E)) ≥ 1 for almost all θ. The conjecture seeks equality, asserting that the visible part typically has full possible dimension 1 in the plane.

The survey reports that this has been verified for certain classes (e.g., graphs of functions, quasi-circles, and typical realizations of Mandelbrot percolation), but the general statement remains a long-standing conjecture.