Kotzig's perfect 1-factorisation conjecture for complete graphs

Let $n\geq 2$ be an even integer. A perfect $1$-factorisation of a graph is a partition of its edge set into perfect matchings such that the union of any two distinct perfect matchings is a Hamilton cycle. Kotzig's conjecture. The complete graph $K_n$ has a perfect $1$-factorisation. Kotzig's conjecture is known when $n-1$ is prime, when $n/2$ is prime, and for some sporadic values; the smallest currently open case is $n=64$.