Bipartite monochromatic path partition conjecture

Let $K_{n,n}$ be a balanced complete bipartite graph whose edges are coloured with $r$ colours. A vertex-partition into monochromatic paths is a partition of all vertices into paths, each having edges of one colour. Bipartite path partition conjecture. Suppose that the edges of $K_{n,n}$ are coloured with $r$ colours. There is a vertex-partition of $K_{n,n}$ into $2r-1$ monochromatic paths. This conjecture would generalise the paper's earlier result for balanced complete bipartite graphs and is proposed as a direction beyond complete graphs.