Extendability conjecture for 5-edge-connected essentially 6-edge-connected graphs

Let $G$ be a graph and let $z_0\in V(G)$. A graph is $\langle\mathbb{Z}_3\rangle$-extendable at $z_0$ if every pre-orientation of the edges incident with $z_0$ whose out-degree minus in-degree at $z_0$ agrees modulo $3$ with a prescribed zero-sum function can be extended to the corresponding $\beta$-orientation of $G$. Extendability conjecture. Every $5$-edge-connected essentially $6$-edge-connected graph is $\langle\mathbb{Z}_3\rangle$-extendable at any vertex of degree $5$. This is presented as a strengthening of the Jaeger–Linial–Payan–Tarsi conjecture, and the paper states that it would imply that conjecture; its resolution is not supplied here.