Almost partitioning every $2$-edge-coloured complete $k$-graph into $k$ monochromatic tight cycles

A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\ge 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$ vertex-disjoint monochromatic tight cycles that together cover $n-o(n)$ vertices.