The unique-counterexample conjecture for locally irregular edge colorings

A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring is locally irregular if every color induces a locally irregular subgraph. A graph is colorable if it admits a locally irregular edge coloring. For a colorable connected graph $G$, let $\chi_{\rm irr}^{\prime}(G)$ be the smallest number of colors required by such a coloring, and let $B$ denote the bow-tie graph. The unique-counterexample conjecture. The bow-tie graph $B$ is the only colorable connected graph with

$$\chi_{\rm irr}^{\prime}(B)>3.$$

The paper proves that $B$ is the only counterexample among cacti, while the proposed assertion for general connected graphs remains open.