Las Vergnas Cube Conjecture and reconstruction properties of the cube matroid

Las Vergnas Cube Conjecture states that the cube matroid has exactly one class of orientations. We prove that this conjecture is equivalent to saying that the oriented matroid of the affine dependencies of the n-cube can be reconstructed from the underlying matroid and one of the following partial lists of signed circuits or cocircuits: A) the signed circuits of rank 3, or B) the positive signed cocircuits.