The conjecture that all cubes are stackable

Let $Q^d$ denote the $d$-dimensional hypercube, and call a graph stackable if, starting with one cup at each vertex, all cups can be moved onto any prescribed target vertex according to the cup-stacking rules. Cube stackability conjecture. For all $d$, $Q^d$ is stackable. The paper proves stackability for cubes through dimension $20$; the conjecture asserts that no dimension imposes an obstruction.