Tight tree-cover conjecture for dense edge-coloured graphs

For $r\geq2$ and $\delta\in(0,1)$, let

$$tc_r(\delta):=\limsup_{n\to\infty}\max_{G\in\mathcal{G}(n,r,\delta)}tc(G),$$

where $\mathcal{G}(n,r,\delta)$ consists of the $n$-vertex $r$-edge-coloured graphs with $\delta(G)\geq(1-\delta)n$, and $tc(G)$ is the smallest number of not necessarily vertex-disjoint monochromatic trees covering $V(G)$. Tight tree-cover conjecture. There exists $K>0$ such that, for all $r\geq2$ and $\delta\in(0,1)$,

$$tc_r(\delta)\leq Kr\left\lceil\frac{r}{\log(1/\delta)}\right\rceil.$$

This is proposed as a stepping stone toward the corresponding cycle-partition conjecture. The source does not state a general resolution; it records stronger results in particular ranges, so the conjecture remains open.