Li et al.'s density conjecture for connected-flow-critical graphs

Let $G$ be a connected-flow-critical graph, meaning that $G$ does not admit a nowhere-zero $3$-flow and, for every non-trivial partition $\mathcal{P}$ of $V(G)$ whose parts each induce a connected graph, the contraction $G/\mathcal{P}$ admits a nowhere-zero $3$-flow. Let $n_3$ denote the number of vertices of degree $3$ in $G$. Li et al.'s density conjecture. For any connected-flow-critical graph $G$ on at least seven vertices with $n_3$ vertices of degree $3$,

$$|E(G)|<\frac{5|V(G)|}{2}+n_3.$$

The conjecture is intended as a density bound for minimal obstructions to nowhere-zero $3$-flows and, unlike the preceding weaker bound, would imply Tutte's $3$-flow conjecture. The source gives no resolution, so it remains open.