The symmetric chain decomposition extension conjecture for hypercubes

A symmetric chain decomposition (SCD) of the hypercube $Q_n$ is a partition of its vertices into symmetric chains, where a symmetric chain is a path $(x_k,x_{k+1},\ldots,x_{n-k})$ such that $x_i$ lies at level $i$ for every $k\leq i\leq n-k$. The symmetric chain decomposition extension conjecture. Every SCD can be extended to a Hamilton cycle in $Q_n$. This conjecture extends results of Streib and Trotter and the Greene–Kleitman construction, which show that particular SCDs can be extended to Hamilton cycles. It is motivated by the conjecture of Ruskey and Savage that every matching in $Q_n$ extends to a Hamilton cycle.