Jain's $S^2$-flow conjecture

Let $G$ be a bridgeless graph, and let $\phi_{3,2}(G)$ be its three-dimensional Euclidean flow index. Jain's $S^2$-flow conjecture.

$$\phi_{3,2}(G)=2$$

Equivalently, every bridgeless graph admits a unit-sphere flow in $\mathbb{R}^3$. The source states that this conjecture, together with Jain's $S^2$-map conjecture, would imply Tutte's $5$-flow conjecture; its resolution is not given.