Lin's conjecture on boundary nodal sets of harmonic functions

Let $\Omega \subset \mathbb R^d$ be a Lipschitz domain and let $\Sigma=B\cap\partial\Omega$ for a ball $B$ centered at a point of $\partial\Omega$. Let $u$ be harmonic in $\Omega$, continuous up to the boundary, and vanish on $\Sigma$. Lin's conjecture. If the set $\{x\in\Sigma:\partial_\nu u(x)=0\}$ has positive surface measure, then $u\equiv 0$. This is a boundary unique-continuation question related to the Bers problem; it remains open in the stated generality, although the analogous planar question has a positive answer and counterexamples are known in higher-dimensional half-spaces for related formulations.