Circular-flow threshold conjecture for two-dimensional normed flows

Let $G$ be a graph and let a circular $(3+\epsilon)$-NZF mean a circular nowhere-zero flow with parameter $3+\epsilon$. Circular-flow threshold conjecture. For $\epsilon\in[0,1]$, there exist two functions $f$ and $g$, with $f(\epsilon)\in[2,+\infty]$ strictly increasing and $g(\epsilon)\in[1,2]$ strictly decreasing, such that if $G$ admits a circular $(3+\epsilon)$-NZF, then

$$\phi_{2,p}(G)=2 \quad \text{for all } p\ge f(\epsilon) \text{ and for all } 1\le p\le g(\epsilon).$$

This conjecture seeks parameter ranges in which a circular-flow hypothesis forces the two-dimensional $p$-normed flow index to attain its minimum; no resolution is supplied.