The 0-sum 3-flow conjecture for 2-edge-connected 5-regular graphs

Let $G$ be a $2$-edge connected $5$-regular graph. A $0$-sum $3$-flow is a nowhere-zero flow whose values lie in $\{-2,-1,1,2\}$ and whose sum at every vertex is zero.

0-sum 3-flow conjecture. Every $2$-edge connected $5$-regular graph admits a $0$-sum $3$-flow.

The preceding theorem establishes the analogous assertion for every odd integer $r\geq 3$ with $r\neq 5$, leaving the $5$-regular case as the exceptional case addressed by this conjecture.