The dual-cube lifting conjecture for completely independent spanning trees

Let $Q_n$ be the $n$-dimensional hypercube and let $F_{n+1}$ be the dual-cube of dimension $n+1$. Let $k$ be a positive integer. Dual-cube lifting conjecture. If $Q_n$ has $k$ completely independent spanning trees, then $F_{n+1}$ also has $k$ completely independent spanning trees. The conjecture is motivated by the fact that each cluster of $F_n$ is isomorphic to $Q_{n-1}$ and that the cluster-connecting graph admits edge-disjoint connecting structures. The paper establishes a construction for two completely independent spanning trees, but the asserted implication for arbitrary $k$ remains open.