The Carleson-measure characterization of multi-nicely connected domains

Let $G$ be a domain conformally equivalent to a circular domain $W$, and let $\phi:G\to W$ be a conformal map. A positive measure on $G$ is a Carleson measure if it satisfies the Carleson embedding condition for the relevant analytic function space. Carleson-measure characterization. If every Carleson measure on $G$ is of the form $\tau\circ\phi$, where $\tau$ is a Carleson measure on $W$, then $G$ is a multi-nicely domain. This gives a converse to the paper's preceding implication relating multi-nicely connected domains and the pullback description of Carleson measures; the source provides no resolution of the converse.