The Local Irregularity Conjecture excluding the bow-tie graph

Let $B$ be the bow-tie graph, and let $G$ be a connected graph that is locally irregularly colorable. Write $\operatorname{lir}(G)$ for its locally irregular chromatic index.

Local Irregularity Conjecture. If $G\neq B$, then

$$\operatorname{lir}(G)\leq 3.$$

This is the improved form of the original conjecture after the bow-tie graph was found to require four colors. The source reports this as a conjecture established by Sedlar and Škrekovski, with the bound proved for cacti other than $B$.