The Hamiltonian-path conjecture for non-conflicting flows

Let $G$ be a bridgeless cubic graph containing a Hamiltonian path, and let $\overline{F}$ denote a $2$-factor of $G$. A non-conflicting nowhere-zero $Z_2\times Z_2$-flow with respect to $\overline{F}$ is the flow notion defined in the paper. The Hamiltonian-path flow conjecture. Unless $G$ is the Petersen graph, there exists a $2$-factor $\overline{F}$ such that $G/\overline{F}$ admits a non-conflicting nowhere-zero $Z_2\times Z_2$-flow with respect to $\overline{F}$. The paper presents this as its final conjecture; it remains open.