This paper considers completions of tope graphs of complexes of oriented matroids (COMs) to ample partial cubes of the same Vapnik--Chervonenkis dimension (VC-dimension). We show that these exist for oriented matroids (OMs) and complexes of uniform oriented matroids (CUOMs). This implies that tope graphs of OMs and CUOMs satisfy the sample compression conjecture---one of the central open questions of learning theory. We conjecture that the tope graph of every COM can be completed to an ample partial cube without increasing the VC-dimension.