Bal–DeBiasio conjecture on monochromatic tree covers

Let $G$ be an $n$-vertex $r$-edge-coloured graph, and let $tc(G)$ be the smallest number of not necessarily vertex-disjoint monochromatic trees whose vertices cover $V(G)$. Bal–DeBiasio conjecture. For every $r\geq2$, if

$$\delta(G)\geq\frac{r(n-r+1)+1}{r+1},$$

then $tc(G)\leq r$. The conjecture was settled affirmatively by Bucić, Korándi, and Sudakov, so it is solved.