The elliptic ordering conjecture for solutions of the outer problem

Let $D$ be a fixed core, and let $\Omega_1$ and $\Omega_2$ be two solutions of the outer problem with respect to $D$. Write $|\Omega|$ for the volume of a domain.

Elliptic ordering conjecture. The solutions of the outer problem form an elliptically ordered family: if $|\Omega_1|<|\Omega_2|$, then

$$\Omega_1\subset \Omega_2.$$

The conjecture concerns global comparison and ordering of solutions with the same core. The paper identifies global existence and uniqueness as difficult and does not provide a proof of this ordering property.