Mutual singularity of harmonic measures from componentwise Carleson measures

Let $G$ be an open subset, with components whose harmonic measures are considered. A positive measure $\mu$ on $G$ is a Carleson measure when it satisfies the relevant Carleson embedding condition, and its restriction to a component means the measure obtained by restricting $\mu$ to that component. Componentwise Carleson-measure conjecture. Suppose that whenever $\mu$ is a Carleson measure on $G$, the restrictions of $\mu$ to each component of $G$ are Carleson measures. Then the harmonic measures of the components of $G$ are mutually singular. This is proposed as a converse to the finite-component corollary stated immediately before it; the source gives no resolution.