The 6-edge-connectivity conjecture for strongly connected modulo 3-orientations

A graph is \mathbb Z_3-connected if, for every function $\beta:V(G)\to\mathbb Z_3$ satisfying $\sum_{v\in V(G)}\beta(v)\equiv 0\pmod{3}$, there is an orientation $D$ of $G$ such that $d_D^+(v)-d_D^-(v)\equiv\beta(v)\pmod{3}$ for every $v\in V(G)$. An orientation is strongly connected if, for every two vertices $x,y\in V(G)$, there is a directed path from $x$ to $y$. A graph is $\mathcal S_3$-connected if, for every $\mathbb Z_3$ boundary function $\beta$, it has a strongly connected orientation $D$ satisfying $d_D^+(v)-d_D^-(v)\equiv\beta(v)\pmod{3}$ for every vertex $v$.

The 6-edge-connectivity conjecture. Every 6-edge-connected graph has a strongly connected modulo $3$-orientation.

This conjecture strengthens the known result that every 6-edge-connected graph is $\mathbb Z_3$-connected. It was proved for 8-edge-connected graphs by Lovász, Thomassen, Wu and Zhang, while the 6-edge-connected case is therefore no longer open.