Khavinson–Shapiro Schwarz potential singularity conjecture

Let $\Gamma$ be a nonsingular analytic surface, and consider the Cauchy problem for Laplace's equation on $\Gamma$ with real-entire data. Let $u$ be its solution, and let $w$ be the Schwarz potential of $\Gamma$. Khavinson–Shapiro conjecture. The singularity set of $u$ is contained in the singularity set of $w$. This conjecture asserts that the Schwarz potential controls the analytic continuation of solutions to the corresponding Cauchy problem; the source states it in the context of entire Schwarz potentials, and notes that it is known in the plane but unresolved in higher dimensions.