The one-phase problem for harmonic measure in two-sided NTA domains

We show that if $\mathrm{\Omega }\subset {\mathbb{R}}^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\text{VMO}(\sigma )$, where $\sigma$ is the surface measure of $\mathrm{\Omega }$, then the outer unit normal to $\mathrm{\partial }\mathrm{\Omega }$ belongs to $\text{VMO}(\sigma )$ too. The analogous result fails for dimensions larger than $3$. This answers a question posed by Kenig and Toro and also by Bortz and Hofmann.