Almost partitioning 2-edge-colourings of 3-uniform hypergraphs with two monochromatic tight cycles

We show that any 2-colouring of the 3-uniform complete hypergraph Kn(3) on n vertices contains two disjoint monochromatic tight cycles of distinct colours covering all but o(n) vertices of Kn(3). The same result holds if we replace tight cycles with loose cycles.