Rainbow forest packing conjecture for globally bounded graphs

Let $\varepsilon>0$. A graph is globally $m$-bounded if no colour is used on more than $m$ edges, and a forest of order $k$ has $k$ edges. Rainbow forest packing conjecture. There exists an $m\in\mathbb{N}$, depending on $\varepsilon$, such that for every $k$, every properly coloured, globally $m$-bounded graph $G$ with $km$ edges has at least $(1-\varepsilon)m$ edge-disjoint rainbow forests of order $k$. This is posed as a question in the concluding remarks and no resolution is supplied, so it remains open.