Gyárfás–Lehel conjecture for monochromatic tree covers

Let $K_{n,m}$ be the complete bipartite graph with parts of sizes $n$ and $m$, and let $\operatorname{tc}_r(K_{n,m})$ be the minimum number of monochromatic trees whose vertices cover $K_{n,m}$ in every edge-colouring with at most $r$ colours. Gyárfás–Lehel conjecture. For all $n,m\ge 1$ and $r\ge 2$,

$$\operatorname{tc}_r(K_{n,m})\le 2r-2.$$

The conjecture is known for $r\le 5$, and examples show that the bound is sharp; the general case remains open.