Converse of unbalanced acyclic orientation and unbalanced coloring

Ascertain whether every graph that admits an unbalanced acyclic orientation necessarily admits an unbalanced coloring of its edges.

Background

The authors show that unbalanced edge colorings lead to unbalanced acyclic orientations via matroidal arguments, while noting uncertainty about the converse direction.

Resolving this equivalence (or finding a counterexample) would clarify the precise relationship between orientation- and coloring-based characterizations that underpin independence in certain rigidity matroids.