Elliptic Schwarz potential conjecture

Let $\alpha$ be an entire function. On a nonsingular analytic surface, let $u$ solve the Cauchy problem for

$$\operatorname{div}(\alpha\nabla u)=0$$

with entire data. Let $q$ solve

$$\operatorname{div}(\alpha\nabla q)=1,$$

and let $v$ solve the corresponding Cauchy problem with data $q$. Elliptic Schwarz potential conjecture. The singularity set of $u$ is contained in the singularity set of $v$. This generalizes the Schwarz potential conjecture from the Laplacian to an elliptic operator with entire coefficient $\alpha$. The source emphasizes that, unlike the classical Schwarz potential conjecture, it is not even known whether this generalized conjecture holds in the plane.