Existence of nonalgebraic quadrature domains in dimensions greater than two

A quadrature domain is understood here in the classical, restricted sense, rather than in the wide sense allowing a compactly supported distribution in the quadrature formula. The claim concerns Euclidean dimensions $n>2$. Nonalgebraic quadrature-domain conjecture. In dimensions greater than two, there exist quadrature domains that are not algebraic. Quadrature domains are well understood in the plane, while in higher dimensions only limited explicit examples and qualitative information are known; in particular, the algebraicity of general higher-dimensional quadrature domains is unresolved.