Electrostatic skeletons and condition of strict descent

Given a precompact domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^2$, the electrostatic skeleton of $\mathrm{\Omega }$ is defined as a positive measure inside $\mathrm{\Omega }$, supported on a set with no simple loops, which generates $\mathrm{\partial }\mathrm{\Omega }$ as an equipotential curve. Eremenko conjectured that every convex polygon admits a unique electrostatic skeleton. This conjecture has since been proven for triangles and regular polygons. In this paper, we will prove the conjecture for quadrilaterals with a line of symmetry using arguments from conformal geometry. We will also discuss a natural condition that implies the existence of electrostatic skeletons.