Perfect 1-factorization conjecture

Let $K_{2n}$ be the complete graph on $2n$ vertices. A perfect matching is a spanning 1-regular subgraph, and a Hamiltonian cycle is a cycle containing every vertex. Perfect 1-factorization conjecture. For every integer $n\geq 2$, $K_{2n}$ can be decomposed into $2n-1$ perfect matchings such that the union of any two matchings forms a Hamiltonian cycle of $K_{2n}$. Perfect 1-factorizations connect edge decompositions with Hamiltonian cycle structure; the source gives no resolution status.