Connectedness suffices for convergence of Faber zero counting measures

Let $G$ be the domain under consideration, let $u_n$ denote the counting measure associated with the $n$th Faber polynomial, and let $bc_L$ be the equilibrium measure of $L$. Convergence conjecture. If $G$ is connected, then

$$\nu_n\overset{*}{\longrightarrow}\mu_L\quad\text{as }n\to\infty.$$

The paper explains that this convergence is known under the stronger assumption A.3 and that a subsequence always converges to $\mu_L$, but whether connectedness alone suffices is left open.