The non-conflicting flow conjecture for 3-edge-connected cubic graphs

Let $G$ be a 3-edge-connected cubic graph different from the Petersen graph. A nowhere-zero $Z_2\times Z_2\times Z_2$-flow is a flow $f$ whose edge values are nonzero elements of $Z_2\times Z_2\times Z_2$. The non-conflicting flow conjecture. There exist $x,y\in Z_2\times Z_2\times Z_2$ such that

1. $f^{-1}(\{x,y\})$ is a matching in $G$;
2. there is no edge $e=uv$ of $G$ for which $u$ is incident with an edge $e_u$ and $v$ is incident with an edge $e_v$ satisfying $f(e_u)=x$ and $f(e_v)=y$. The paper states that this conjecture implies the normal chromatic-index bound $\chi'_{N}(G)\leq 6$ for bridgeless cubic graphs; it remains open.