Combinatorial proof of the unbalanced coloring ⇒ unbalanced acyclic orientation implication

Provide a combinatorial proof that any graph with an unbalanced edge coloring (no monochromatic cycles or alternating color trails) necessarily admits an unbalanced acyclic orientation.

Background

The paper extends a characteristic-independent argument showing that an unbalanced coloring implies independence in a certain matroid, which in turn implies existence of an unbalanced acyclic orientation. However, the authors note the lack of a purely combinatorial proof of this implication.

A combinatorial proof would sharpen understanding of the relationship between coloring-based and orientation-based characterizations in rigidity-related matroids, particularly hyperconnectivity and wedge-power contexts.