Baranyai–Katona wreath conjecture

Let $k\leq n$ be positive integers. Write $\mathbb{Z}_{n}$ for the integers modulo $n$, and let $\mathbb{Z}_{n}^{(k)}$ be the set of $k$-element subsets of $\mathbb{Z}_{n}$. A wreath is a set of the form $\mathcal{F}_{\pi}$ obtained from a permutation $\pi$ of $\mathbb{Z}_{n}$ by taking the cyclically indexed blocks of $k$ consecutive values, as defined in the paper.

The wreath conjecture. For any positive integers $k\leq n$, there is a decomposition of $\mathbb{Z}_{n}^{(k)}$ into disjoint wreaths.

This conjecture generalizes Baranyai's factorization theorem for complete uniform hypergraphs. It was posed by Baranyai and reformulated in terms of wreaths by Katona; the source does not state a resolution.