The flow conjecture for 3-edge-connected cubic graphs excluding the Petersen graph

Let $G$ be a 3-edge-connected cubic graph different from the Petersen graph $P_{10}$. A nowhere-zero $\mathbb{Z}_2^3$-flow is a flow $f:E(G)\to\mathbb{Z}_2^3\setminus\{0\}$ satisfying flow conservation at every vertex. Flow conjecture for 3-edge-connected cubic graphs excluding $P_{10}$. There exist $\alpha,\beta\in\mathbb{Z}_2^3$ such that

$$f^{-1}(\{\alpha,\beta\})$$

is a matching in $G$, and there is no edge $e=uv$ of $G$ for which $u$ is incident to an edge $e_u$ and $v$ is incident to an edge $e_v$ satisfying $f(e_u)=\alpha$ and $f(e_v)=\beta$. If true, this would provide the flow-based route proposed in the paper toward the normal 6-edge-coloring conjecture. No resolution is supplied here.