The dihedral 2-flow conjecture for 3-edge-colorable cubic graphs

Let $\Gamma$ be a $3$-edge-colorable cubic graph. An embedding of $\Gamma$ in a surface is an embedding whose genus is the genus of $\Gamma$, and a nowhere-identity dihedral $2$-flow is a flow with values in the dihedral group of order $4$ such that no edge receives the identity element. Dihedral 2-flow conjecture. For every $3$-edge-colorable cubic graph, there exists an embedding in a surface of its own genus, with respect to which it has a nowhere-identity dihedral $2$-flow. Confirmation would imply a positive answer to whether every planar cubic graph has a nowhere-identity dihedral $3$-flow; the source gives no resolution of this conjecture.