The golden inequality characterization of planarity for cubic graphs

Let $G$ be a cubic bridgeless graph with $E$ edges, let $F_G(Q)$ denote its flow polynomial, and let $\phi=(1+\sqrt{5})/2$ be the golden ratio. Golden inequality conjecture.

$$(-\phi)^E F_G((5-\sqrt{5})/2) \geq F_G((3-\sqrt{5})/2)^2.$$

Moreover, $G$ is planar if and only if equality holds. This conjecture asserts that the Tutte golden identity for the flow polynomial characterizes planarity among cubic bridgeless graphs. It is known for near-planar graphs, but remains open in general.