Partitioning $2$-coloured complete $k$-uniform hypergraphs into monochromatic $\mathrm{\ell }$-cycles

We show that for all $\mathrm{\ell },k,n$ with $\mathrm{\ell }\le k\mathrm{/}2$ and $(k-\mathrm{\ell })$ dividing $n$ the following hypergraph-variant of Lehel’s conjecture is true. Every $2$-edge-colouring of the $k$-uniform complete hypergraph ${\mathcal{K}}_n^{(k)}$ on $n$ vertices has at most two disjoint monochromatic $\mathrm{\ell }$-cycles in different colours that together cover all but at most $4(k-\mathrm{\ell })$ vertices. If $\mathrm{\ell }\le k\mathrm{/}3$, then at most two $\mathrm{\ell }$-cycles cover all but at most $2(k-\mathrm{\ell })$ vertices. Furthermore, we can cover all vertices with at most $4$ ($3$ if $\mathrm{\ell }\le k\mathrm{/}3$) disjoint monochromatic $\mathrm{\ell }$-cycles.