Distinct-color monochromatic partition and cover conjecture

Let $G$ be a graph on $n$ vertices and let $r\geq 1$. Write $\mathcal{T}\mathcal{C}_r$ for the property that every $r$-edge-coloring of $G$ admits a cover by at most $r$ monochromatic connected subgraphs of distinct colors, and write $\mathcal{T}\mathcal{P}_r$ for the analogous partition property.

Distinct-color conjecture. If

$$\delta(G)\geq \left(1-\frac{1}{2^r}\right)n,$$

then $G$ has property $\mathcal{T}\mathcal{C}_r$ (and property $\mathcal{T}\mathcal{P}_r$).

The source notes that the statement is known for $r=1$ and $r=2$, and presents the general assertion as open; the cited example indicates that the minimum-degree bound would be best possible.