The planar graph equilibrium conjecture

Let $G$ be a strongly connected planar graph, and let $g$ denote its undirected girth. A static equilibrium is an equilibrium of the static type, while a $2$-chase equilibrium and a WT equilibrium are the cycle-based equilibrium types defined in the paper. Planar graph equilibrium conjecture. All strongly connected planar graphs with

$$g\geq 4$$

have either a static equilibrium, a $2$-chase equilibrium, or a WT equilibrium. The conjecture extends the paper's result for strongly connected outerplanar graphs and is motivated by constructions of strongly connected graphs of girth $5$ without cycle-based equilibria, all of which are non-planar; its resolution for general planar graphs remains open.