The degree-sensitive density conjecture for 3-flow-critical graphs

Let $G$ be a 3-flow-critical graph, meaning that $G$ is bridgeless, does not admit a nowhere-zero $3$-flow, and $G/e$ admits one for every $e\in E(G)$. Let $n=|V(G)|$, let $E(G)$ be its edge set, and let $V_3(G)$ be the set of vertices of degree $3$; write $n_3=|V_3(G)|$.

Degree-sensitive density conjecture. For every $3$-flow-critical graph $G$ on $n$ vertices,

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

The authors propose this as a stronger conjecture than the preceding upper-density conjecture and state that it would imply Tutte's $3$-flow conjecture. It is presented as open in the paper.