The zero-area hitting measure conjecture for random planar curves

Let $\gamma$ be an unbounded one-sided curve in the Euclidean plane, and let $\Omega$ be a simply connected, open, bounded planar domain. Reroot $\gamma$ at a root chosen in $\Omega$ and rotate it about its root by an independent angle uniformly distributed in $[0,2\pi]$. The resulting random curve has a hitting-point distribution on $\partial\Omega$. Zero-area hitting measure conjecture. For any $\gamma$ and $\Omega$, for almost every root, the corresponding measure on $\partial\Omega$ has $0$ two-dimensional Lebesgue measure. This concerns the possible singularity of hitting measures for random translations and rotations of planar curves; the supplied text presents it as a problem, and no resolution is given here.