Monochromatic components in dense 2-edge-coloured balanced bipartite graphs

We prove that each $2$-edge-coloured spanning subgraph $G$ of $K_{n,n}$ with $\delta (G)\ge \lfloor (2n+1)\mathrm{/}3\rfloor$ can be covered by at most three monochromatic components. We provide a $2$-edge-coloured spanning subgraph of $K_{n,n}$ showing this minimum degree condition is sharp.