Jaeger–Linial–Payan–Tarsi conjecture on _3-connected graphs

Let $G$ be a finite graph without loops, possibly with multiple edges. A graph is $\mathbb{Z}_3$-connected if, for every zero-sum function $\beta:V(G)\to\mathbb{Z}_3$, there is an orientation $D$ such that

$$d_D^+(v)-d_D^-(v)=\beta(v)\pmod 3$$

for every vertex $v\in V(G)$. Jaeger, Linial, Payan and Tarsi's conjecture. Every $5$-edge-connected graph is $\mathbb{Z}_3$-connected. This generalizes the relationship between nowhere-zero flows and modulo orientations; it remains open, while every $6$-edge-connected graph is known to be $\mathbb{Z}_3$-connected.