The four-color locally irregular decomposition conjecture

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

$${\rm lir}(G)\leq 4.$$

The conjecture was proposed after Sedlar and Škrekovski showed that the bow-tie graph is not decomposable into three locally irregular graphs. It weakens the earlier bound-three conjecture and remains open in the generality stated.