  We prove a contact non-squeezing result for a class of embeddings between
starshaped domains in the contactization of the symplectization of the unit
cotangent bundle of certain manifolds. The class of embeddings includes
embeddings which are not isotopic to the identity. This yields a new proof that
there is no positive loop of contactomorphisms in the unit cotangent bundles
under consideration. The proof uses the shape invariant introduced by Sikorav
and Eliashberg.
