Unique continuation conjecture for harmonic functions in Lipschitz domains

Let $\Omega\subset\mathbb{R}^n$ be a Lipschitz domain and let $\Sigma\subset\partial\Omega$ be relatively open with respect to $\partial\Omega$. Let $u$ be harmonic in $\Omega$ and continuous on $\overline\Omega$. Suppose that $u$ vanishes on $\Sigma$ and that the normal derivative $\partial_\nu u$ vanishes on a subset of $\Sigma$ with positive surface measure. Unique continuation conjecture. Then $u\equiv 0$ on $\overline\Omega$. This boundary unique-continuation problem concerns whether simultaneous vanishing of a harmonic function and its normal derivative on a boundary set of positive surface measure forces the function to vanish identically. It is open for general Lipschitz domains; the paper proves related results for $C^1$ domains and Lipschitz domains with small local Lipschitz constant.