Strongly connected modulo 3-orientations in 6-edge-connected graphs

Prove that every 6-edge-connected graph has a strongly connected modulo 3-orientation, thereby establishing the conjecture that all such graphs have flow index strictly less than three.

Background

A graph is S3-connected if every Z3-boundary function can be realized by a strongly connected orientation whose outdegree-minus-indegree at each vertex agrees with the prescribed boundary value modulo 3. The existence of a strongly connected modulo 3-orientation is equivalent to having flow index strictly less than three.

The cited prior work established that every 8-edge-connected graph has a strongly connected modulo 3-orientation and verified the assertion for several graph families, but the 6-edge-connected case remains unresolved. The paper proves a complete characterization of graphic degree sequences admitting S3-connected realizations and presents this result as support for the conjecture, rather than proving the conjecture for all 6-edge-connected graphs.