Ryser–Lovász conjecture for monochromatic tree covers

Let $G$ be a graph, let $r\text{-} obreak\text{colorings}$ denote edge-colorings of $G$ with $r$ colors, and let $\mathrm{tc}_r(G)$ be the minimum number of monochromatic connected subgraphs whose vertex sets cover $V(G)$ in every such coloring. Let $alpha(G)$ be the independence number of $G$.

Ryser–Lovász conjecture. For every integer $r\geq 2$ and every graph $G$,

$$\mathrm{tc}_r(G)\leq (r-1)\alpha(G).$$

The conjecture is best possible when $r-1$ is a prime power. It is known for $r=2,3$ and for complete graphs when $r\leq 5$, but remains open for $r\geq 4$ in general.