Schwarz potential conjecture for a three-dimensional Neumann oval

Let $a>b>c>0$, and define

$$\Omega:=\{(x,y,z)\in\mathbb{R}^3:(x^2+y^2+z^2)^2-(a^2x^2+b^2y^2+c^2z^2)<0\}.$$

Let $\Gamma$ be the boundary of $\Omega$, and let $W({\bf x})$ be its Schwarz potential. Define

$$B:=\left\{z=0,(x^2+y^2)^2-\frac{x^2}{4(a^2-c^2)}-\frac{y^2}{4(b^2-c^2)}<0\right\}.$$

Three-dimensional Neumann-oval conjecture. $W$ can be analytically continued throughout $\Omega\setminus B$. This is a proposed higher-dimensional analogue of the Schwarz potential behavior for Neumann ovals: the singularity set is conjectured to be confined to a subset of the $xy$-plane bounded by a two-dimensional Neumann oval. The source presents this as an expected result for a degree-four surface and does not provide a proof.