Milićević's bounded-diameter conjecture for monochromatic component covers

Let $r$ be a positive integer. An $r$-edge-coloured complete graph is a complete graph whose edges receive one of $r$ colours, and the diameter of a component is measured in its monochromatic subgraph.

Milićević's conjecture. For every $r$, there is a constant $D(r)$ such that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic components of diameter at most $D(r)$.

This strengthens the complete-graph case of Ryser's conjecture. The source reports that it is proved for $r\leq 4$, with bounds $D(3)\leq 4$ and $D(4)\leq 6$, while the general case remains open.