Proof of the planar Khavinson-Shapiro conjecture

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded domain whose boundary is a finite union of pairwise disjoint Jordan curves. We prove the planar Khavinson–Shapiro conjecture in this setting: if every polynomial on ${\mathbb{R}}^2$ agrees on $\mathrm{\partial }\mathrm{\Omega }$ with a harmonic polynomial, then $\mathrm{\partial }\mathrm{\Omega }$ is an ellipse and $\mathrm{\Omega }$ is its bounded interior.