A counterexample regarding a two-phase problem for harmonic measure in VMO

Let ${\mathrm{\Omega }}^{+}\subset {\mathbb{R}}^{n+1}$ be a vanishing Reifenberg flat domain such that ${\mathrm{\Omega }}^{+}$ and ${\mathrm{\Omega }}^{-}={\mathbb{R}}^{n+1}\setminus \overline{{\mathrm{\Omega }}^{+}}$ have joint big pieces of chord-arc subdomains and the outer unit normal to ${\mathrm{\Omega }}^{+}$ belongs to $VMO({\omega }^{+})$, where ${\omega }^{\pm }$ is the harmonic measure of ${\mathrm{\Omega }}^{\pm }$. Up to now it was an open question if these conditions imply that $\log {\displaystyle \frac{d{\omega }^{-}}{d{\omega }^{+}}}\in VMO({\omega }^{+})$. In this paper we answer this question in the negative by constructing an appropriate counterexample in ${\mathbb{R}}^2$, with the additional property that the outer unit normal to ${\mathrm{\Omega }}^{+}$ is constant ${\omega }^{+}$-a.e. in $\mathrm{\partial }{\mathrm{\Omega }}^{+}$.