The bounded-color monochromatic-cover conjecture

Let $r\geq 2$ and let $K$ be a complete graph whose edges are colored with colors in $[r]$. A monochromatic cover is a collection of monochromatic connected subgraphs covering all vertices, and its order is its number of subgraphs.

Bounded-color cover conjecture. Every such coloring has a monochromatic cover of order at most $r-1$ in which the subgraphs use at most $\lceil r/2\rceil$ distinct colors.

This is the balanced-subset special case of the S-versus-complement conjecture.