On comparing systems of random variables with the Rademacher sequence

We ask whether inequalities between distributions of scalar polynomials of two sequences of random variables imply that the corresponding inequalities hold between the distributions of the norms of the corresponding vector sums in an arbitrary Banach space provided that one of the systems is the Rademacher system. We show that the answer is affirmative when the Rademacher functions form the majorizing system, and negative in the opposite case.