Analytic-boundary alternative for nice harmonic-function quotients

Let $\Omega$ be a one-sided NTA domain, and let $u$ and $v$ be positive harmonic functions as above, vanishing on the relevant portion of the boundary, such that their quotient agrees on $B(z,r)\cap\partial\Omega$ with a real-analytic function $R$. Analytic-boundary alternative. 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 dimension $n-2$. This is the first and most difficult of several proposed implications connecting analytic boundary data for harmonic functions with regularity of the boundary; its resolution is not given here.