A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain.

I have found a claim in a preprint from 2008 that no such theorem is known for cubical complexes. Can anyone substantiate or disprove this claim?

projectionsof the full cube--a very restricted notion but sufficiently flexible for the homotopy & homology. $\endgroup$ – Włodzimierz Holsztyński Oct 7 '16 at 17:04