Nontrivial summable ideal below a cotype-generated representation

Determine whether a B-representation in a Banach space of non-trivial cotype can be chosen so that the associated summable ideal defined by the q-th powers of the vector norms is non-trivial.

Background

The paper proves that if a B-ideal is represented in a Banach space with non-trivial cotype q, then it is contained in the summable ideal Sum((||x_n||^q)_n). The authors leave unresolved whether the representation can be selected so that this containing summable ideal is itself non-trivial.