Gyárfás–Lehel biclique monochromatic-component conjecture

Let $G$ be a biclique, meaning a complete bipartite graph, whose edges are colored with $r$ colors. A monochromatic component is a connected component of the subgraph formed by the edges of one color.

Gyárfás–Lehel conjecture. In every $r$-coloring of the edges of $G$, the vertex set can be covered by the vertices of at most $2r-2$ monochromatic components.

This is the bipartite analogue of the complete-graph monochromatic-component conjecture. The source notes that the bound is sharp, but does not state a resolution of the conjecture.