Khavinson–Shapiro conjecture for planar domains

Let $\Omega\subset\mathbb{C}$ be a bounded domain whose boundary consists of finitely many non-intersecting Jordan curves. For each polynomial boundary datum on $\partial\Omega$, consider the solution of the Dirichlet problem on $\Omega$, namely a function harmonic on $\Omega$, continuous on $\overline{\Omega}$, and equal to the datum on $\partial\Omega$. Khavinson–Shapiro conjecture. If the solution of the Dirichlet problem on $\Omega$ with every polynomial datum is again polynomial, then $\Omega$ must be an ellipse. This is known for ellipses, but the conjecture remains open in general; the paper proves it under an additional algebraic-harmonic-function hypothesis.