Proportionality conjecture for regular Cantor boundaries

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 $R=|A|$ for a nonconstant holomorphic function $A$ on $B(z,r)$. In addition, suppose that $\partial\Omega$ is a regular Cantor set: there are $a>1$, $\delta>0$, and $C$ such that for every $k\in\mathbb Z_+$, the $a^{-k}$-neighborhood of $\partial\Omega$ is a union $V_1\cup\cdots\cup V_{m_k}$ with $\operatorname{length}(V_i)\leq Ca^{-k}$ and $m_k\leq Ca^{\delta k}$. Proportionality conjecture. Then $u=\lambda v$ for some constant $\lambda$. The claim expresses that a Cantor boundary cannot have the real-analytic alternative from the preceding conjecture; no proof or resolution is supplied here.