The three-color conjecture for locally irregular chromatic index

Let $G$ be a finite simple decomposable graph, meaning that it admits a locally irregular edge-coloring. Its locally irregular chromatic index, denoted by $\chi_{\mathrm{irr}}'(G)$, is the smallest number of colors in such a coloring. Baudon–Bensmail–Przybyło–Wozniak's three-color conjecture. Every decomposable graph $G$ satisfies

$$\chi_{\mathrm{irr}}'(G) \le 3.$$

The conjecture asserts a universal three-color bound for locally irregular edge-colorings. The paper improves known general upper bounds but does not resolve this conjecture.