The corrected Local Irregularity Conjecture excluding the bow-tie graph

Let $G$ be a connected locally irregular colorable graph, let $B$ be the bow-tie graph, and let $\mathrm{lir}(G)$ denote the minimum number of locally irregular subgraphs whose edge-disjoint union is $G$. A graph is locally irregular if adjacent vertices have different degrees. Corrected Local Irregularity Conjecture. If $G\neq B$, then

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

The bow-tie graph is the unique counterexample to the original conjecture among colorable cacti, with locally irregular chromatic index four. The corrected statement is presented as the strengthened form motivated by that counterexample; its general status is not resolved in the supplied text.