The bow-tie exception conjecture for locally irregular decompositions

Let $\mathfrak{T'}$ be the family consisting of the recursively defined family $\mathfrak{T}$, all odd-length paths and all odd-length cycles, and let $B$ denote the bow-tie graph. For a graph $G$, let ${\rm lir}(G)$ be the least number of locally irregular graphs into which $G$ can be decomposed. Bow-tie exception conjecture. Every connected graph $G\notin\mathfrak{T'}$, except for the bow-tie graph $B$, satisfies

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

The bow-tie graph is known not to admit a decomposition into three locally irregular graphs, while the conjecture has been proved for trees, cacti, sufficiently high minimum degree, and sufficiently large regular degree. It remains open for general connected graphs.