Rich flow number conjecture

Let $G$ be a graph with maximum degree $\Delta$. A rich $k$-flow is a nowhere-zero $k$-flow $\phi$ such that $|\phi(e)|\neq |\phi(f)|$ for every pair of adjacent edges $e$ and $f$. The graph $G$ is rich flow admissible if it is bridgeless and no two edges forming a $2$-edge-cut are incident with a common vertex.

Rich flow number conjecture. If $\Delta\geq 5$, then $G$ admits a rich $\lfloor 1.5\Delta+1\rfloor$-flow.

The paper proves a linear upper bound of $(264\Delta-521)$ for rich flow admissible graphs. The conjectured bound is motivated by examples whose rich flow number is $3k+1$ when the maximum degree is $2k$, and remains open.