Finite-genus classification conjecture for planar exceptional domains

Let an exceptional domain in $\mathbb{R}^2$ be a domain supporting the overdetermined potential-theoretic problem considered in the paper, and let its genus be finite. The three examples under consideration are the exterior of the unit disk, a halfplane, and the nontrivial domain described in the paper's theorem. Finite-genus classification conjecture. The only finite genus exceptional domains in $\mathbb{R}^2$ are the exterior of the unit disk, the halfplane, and the domain described in the paper's theorem. The conjecture is motivated by the existence of infinite-genus examples and is confirmed in the paper's discussion for domains with finitely many boundary components, while the full finite-genus classification remains open.