Boundary unique continuation conjecture for harmonic functions on Lipschitz domains

Let $\Omega$ be a Lipschitz domain, and let $u\in H^1$ be a weak solution to the Dirichlet problem

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Suppose that the set $\{\partial u/\partial\boldsymbol{n}=0\}\cap\partial\Omega$ has positive surface measure. Boundary unique continuation conjecture. Then $u\equiv 0$. This is a long-standing conjecture concerning boundary asymptotics and unique continuation for harmonic functions near conical points; its resolution is not indicated here.