Chen–Fujita–Gyárfás–Lehel–Tóth bipartite tree-cover conjecture

Let $K_{n,m}$ be a complete bipartite graph, and let $\operatorname{tc}_{r,1}(K_{n,m})$ denote the minimum number of monochromatic trees needed to cover every $r$-colouring of its edges. Chen–Fujita–Gyárfás–Lehel–Tóth conjecture. If $r>1$, then

$$\operatorname{tc}_{r,1}(K_{n,m})\leq 2r-2$$

for all $n,m\geq1$. This is the complete-bipartite analogue of the tree-cover problem, and the source presents the bound as a conjecture without giving a resolution.