Planar Circular Flow Conjecture

Let $k$ be a positive integer, and let $G$ be a planar graph. A circular $\frac{p}{q}$-flow of $G$ is a flow $(D,f)$ such that every edge $e\in E(G)$ satisfies

$$q\leqslant |f(e)|\leqslant p-q.$$

Planar Circular Flow Conjecture. Every $4k$-edge-connected planar graph admits a circular $(2+\frac{1}{k})$-flow.

This conjecture is motivated by Jaeger's conjecture that every $4k$-edge-connected graph admits a circular $(2+\frac{1}{k})$-flow, which was disproved for all $k\geqslant 3$ by Han, Li, Wu, and Zhang. The cited counterexamples are nonplanar, so the planar case remains the relevant unresolved question.