Set-coloured complete-graph tree-cover conjecture

Let $K_n$ be the complete graph on $n$ vertices. In an $(r,k)$-colouring, each edge receives a $k$-element subset of a set of $r$ colours; let $\operatorname{tc}_{r,k}(K_n)$ be the minimum number of monochromatic trees needed to cover every such colouring. Set-coloured tree-cover conjecture. For all $n\geq 1$ and $r>k\geq 1$,

$$\operatorname{tc}_{r,k}(K_n)\leq r-k.$$

The claim follows from Gyárfás's conjecture if that conjecture holds, and is known in several ranges, including $k>r/2$ and $k\geq r/2-1$; later work cited in the source proves it for $k>r/4$. It is not tight in general, as shown by $\operatorname{tc}_{5,2}(K_n)=2$ for $n\geq4$, and remains open in full generality.