The density conjecture for \mathbb{Z}_3-flow-critical graphs

A graph $G$ is $\mathbb{Z}_3$-flow-critical if it is flow-critical with respect to nowhere-zero $\mathbb{Z}_3$-flows, as defined in the source. Density conjecture for $\mathbb{Z}_3$-flow-critical graphs. Every $\mathbb{Z}_3$-flow-critical graph $G$ satisfies

$$|E(G)|\leq 3|V(G)|-5.$$

Moreover, if $|V(G)|\geq 7$, then

$$|E(G)|\leq 3|V(G)|-8.$$

This conjecture concerns the extremal density of nonplanar $\mathbb{Z}_3$-flow-critical graphs. The stated examples $K_{3,n-3}^+$ attain the second bound, while the paper proves related upper bounds and establishes the conjecture in several cases; the general assertion remains open.