Lund's conjecture on non-algebraic quadrature domains

Let $n>2$. A quadrature domain in $\mathbb{R}^n$ is a domain satisfying a quadrature identity for the relevant class of functions, and an algebraic domain is one whose boundary is contained in the zero-set of a polynomial. Lund's conjecture. In all dimensions greater than two, there exist quadrature domains that are not algebraic. This conjecture asks whether the algebraicity of planar quadrature domains extends to higher dimensions; the paper states that it is confirmed in dimension $n=4$, while the claim in all dimensions greater than two is presented here as the conjecture being addressed.