Fan's weighted refinement of the 4-flow conjecture

Let $G$ be a graph with a circuit $C$, and let $\omega:E(G)\to\mathbb{Z}$ be an integer edge-weight function. A $\mathbb{Z}_2\times\mathbb{Z}_2$-flow is a flow with values in the abelian group $\mathbb{Z}_2\times\mathbb{Z}_2$; for such a flow $h$, write $E_{h=(0,0)}(C)$ for the edges of $C$ on which $h$ has value $(0,0)$, and write $\omega(S)=\sum_{e\in S}\omega(e)$.

Fan's weighted refinement. If there is a $\mathbb{Z}_2\times\mathbb{Z}_2$-flow $f$ such that

$$E(G)-E(C)\subseteq \operatorname{supp}(f),$$

then there is a $\mathbb{Z}_2\times\mathbb{Z}_2$-flow $g$ in $G$ such that

$$E(G)-E(C)\subseteq \operatorname{supp}(g)$$

and

$$\omega\bigl(E_{g=(0,0)}(C)\bigr)<\frac{1}{4}\omega(C).$$

This is presented as a refinement of Fan's conjecture and would control the weighted contribution of zero-valued edges on the circuit. Its resolution is not given in the supplied text.