Planar analytic-boundary conjecture for harmonic-function quotients

Let $n=2$. Let $\Omega$ be a one-sided NTA domain, and let $u$ and $v$ be as above, with $u/v$ agreeing on $B(z,r)\cap\partial\Omega$ with a real-analytic function $R$. Assume moreover that $R=|A|$, where $A$ is a nonconstant holomorphic function on $B(z,r)$. Planar analytic-boundary conjecture. Either $u=\lambda v$ for some constant $\lambda$, or $B(z,r)\cap\partial\Omega$ is real analytic, possibly with the exception of a set of isolated points. This conjecture is proved for simply connected $\Omega$ by the results attributed in the source to [VaVo] and [VaVo1], while the infinitely connected case remains to be eliminated.