One-component conjecture for the complement in dumbbell domains

Let $\Omega_h$ be a dumbbell domain, let $D=D_{\alpha,A}$ be an optimal configuration, and let $\psi_{\Omega}$ be the function whose maxima determine the relevant localization regions. One-component conjecture for dumbbells. For every $\alpha>0$, there is $\rho_0(\alpha,h)>0$ such that $D^c$ consists of one component, near one of the maxima of $\psi_{\Omega}$, whenever $|\Omega|-A<\rho_0(\alpha,h)$. This is motivated by numerical evidence and remains open; the authors expect $\rho_0(\alpha,h)\to0$ as $\alpha\to0$.