Subcubic decomposable graph conjecture on the locally irregular chromatic index

A graph is decomposable if it admits a locally irregular edge-coloring (LIEC), and let $\chi_{\mathrm{irr}}'(G)$ denote the smallest number of colors in a LIEC of a decomposable graph $G$. A graph is subcubic when its maximum degree is at most $3$. Subcubic decomposable graph conjecture. For every decomposable graph $G$ with maximum degree $3$,

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

The paper proves the bound for several classes of decomposable subcubic graphs, but many decomposable subcubic graphs remain untreated; the conjecture is presented as a weaker version of the refuted universal 3-color conjecture.