Let X be a smooth manifold of dimension d and M an oriented submanifold of dimension p $\lt$ d which (absolutely) minimizes p-volume in its integral homology class. Is M calibrated by some p-form w?

Definitions: A p-form w is called a calibration if it is closed and its evaluation on every geometric p-vector v of norm one is at most one in norm, |w(v)| $\le$ 1. One says a p-dimensional submanifold M is calibrated (by a calibration w) if w evaluates to 1 on each unit tangent p-vector to M. The word "geometric" above is used to distinguish primitive (or geometric) p-vectors from linear combinations of these; "geometric" means "rank one" in tensor language.

Note: It is elementary that if M is calibrated by any w then M minimizes p-volume in its homology class, so I'm asking for a kind of converse. In the case p+1=d the converse amounts to a continuum version of Max Flow / Min Cut which is discussed in John Sullivan's 1990 Princeton Ph.D. thesis. I tried asking a form of this question on MO on July 24, 2014, but I did not use the term "calibration". I'm hoping with the correct language an expert will notice the question and be able to answer.