Four-colour counterexamples for monochromatic path partitions

In 1989, Gyárfás conjectured that the vertex set of every r-edge-coloured complete graph can be partitioned into at most r monochromatic paths. The case $r=2$ follows from a theorem of Gerencsér and Gyárfás, and Pokrovskiy proved the conjecture for $r=3$. Recently, Li and Wang disproved the conjecture for all sufficiently large r, leaving open the smallest number of colours for which it fails. We show that this number is four. For every integer n≥118, we construct a four-edge-colouring of Kₙ in which every monochromatic path partition contains at least five paths. We also construct, for every integer n≥60, a four-edge-colouring of Kn,n in which every monochromatic path partition contains at least eight paths, disproving Pokrovskiy's conjectured bound $2r-1$ for balanced complete bipartite graphs when $r=4$.