Eremenko's conjecture is proven for convex quadrilaterals with a line of symmetry: they admit a unique electrostatic skeleton supported on a loop-free set.
Saff and Vilmos Totik.Logarithmic potentials with external fields, volume 316 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]
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Electrostatic skeletons and condition of strict descent
Eremenko's conjecture is proven for convex quadrilaterals with a line of symmetry: they admit a unique electrostatic skeleton supported on a loop-free set.