The exceptional-order conjecture for decomposing complete graphs into squares of Hamilton cycles

Let $K_n$ be the complete graph on $n$ vertices, let $C_n$ be a Hamilton cycle on these vertices, and let $C_n^2$ denote its square, obtained by joining pairs at distance at most two on the cycle. A square-of-Hamilton-cycle decomposition conjecture. For any $n \equiv 1 \pmod 4$, $n \not= 9$, there is a decomposition of $E(K_n)$ into copies of $C_n^2$. This strengthens the expected divisibility condition for packing squares of Hamilton cycles and is intended to guide explicit constructions; the cases $n=21,25,33,45,49,57,65,69$ are known, while $n=73$ is described as the smallest unresolved case.