Hung's Hamiltonian decomposition conjecture for augmented cubes

An $n$-dimensional augmented cube is a graph in which each vertex is represented by an $n$-bit binary string, with adjacency defined according to the augmented-cube construction. A Hamiltonian decomposition is a collection of $\lfloor n/2\rfloor$ pairwise edge-disjoint Hamiltonian cycles whose edges partition the edge set.

Hung's conjecture. An $n$-dimensional augmented cube admits a Hamiltonian decomposition.

Hypercubes admit Hamiltonian decompositions, and the conjecture asks whether the analogous property holds for every augmented cube. As far as the source indicates, this conjecture remains open.