Two-color conjecture for strong local irregularity

Prove that every digraph has strong locally irregular chromatic index at most two, where adjacent vertices in each color class must have different balanced degrees.

Background

Strong local irregularity distinguishes adjacent vertices by balanced degree, defined as outdegree minus indegree. The paper proves the two-color bound for symmetric digraphs, Eulerian digraphs, and several cactus subclasses, but leaves the conjecture unresolved for arbitrary digraphs. The concluding discussion also identifies obtaining any general constant upper bound as an open problem.