The bounded-diameter bipartite monochromatic-cover conjecture

Let $\mathcal{K}_2$ be the family of complete bipartite graphs. A monochromatic cover is a collection of monochromatic connected subgraphs covering all vertices, and its order is its number of subgraphs.

Bipartite bounded-diameter conjecture. There is an integer $d$ such that, for every $r\geq 2$, every $r$-coloring of every $K\in\mathcal{K}_2$ has a monochromatic cover of order at most $2r-2$ whose subgraphs all have diameter at most $d$.

The paper proves this for $r=2$ and $r=3;$ the general statement remains open.