The monochromatic path partition conjecture for infinite complete bipartite graphs

Suppose the edges of an infinite complete bipartite graph are coloured with $rin \omega$ colours. Then its vertices can be partitioned into $2r-1$ disjoint monochromatic paths.

Infinite bipartite monochromatic path conjecture. The vertices can be partitioned into $2r-1$ disjoint monochromatic paths.

This asks whether the finite-colour path-decomposition result extends from infinite complete graphs to infinite complete bipartite graphs; the statement is presented as an open problem.