The 3m conjecture for irregularising walks

Let $G$ be a nice graph of size $m$, and let ${\rm ML}^{\rm W}(G)$ denote the minimum size of an irregularising walk of $G$. The 3m conjecture for irregularising walks states that

$${\rm ML}^{\rm W}(G) \leq 3m.$$

The paper proves the general upper bound ${\rm ML}^{\rm W}(G)\leq 4m$ and gives examples with parameter about $2m$; it conjectures that the factor $3$ is sufficient for every nice graph, motivated by subdivided stars whose parameter is expected to approach $3m$.