The bow-tie version of the Local Irregularity Conjecture

Let $G$ be a connected graph, let $\mathfrak{T'}$ be the family consisting of the recursively defined family $\mathfrak{T}$ together with all odd-length paths and odd-length cycles, and let $B$ be the bow-tie graph. Let ${\rm lir}(G)$ denote the locally irregular chromatic index of a locally irregular colorable graph $G$. Bow-tie version of the Local Irregularity Conjecture. Every connected graph $G\notin\mathfrak{T'}$, except for the bow-tie graph $B$, satisfies

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

This formulation incorporates the bow-tie counterexample to the original conjecture. The cited work proves the bound for every locally irregular colorable cactus other than $B$, but the general assertion remains unresolved in the supplied text.