Nontrivial summable ideal under cotype representations

Determine whether a B-ideal represented in a Banach space with non-trivial cotype can be represented by a sequence whose associated summable ideal is non-trivial.

Background

The paper proves that if a sequence represents a B-ideal in a space with non-trivial cotype q, then the B-ideal is contained in the summable ideal generated by the q-th powers of the sequence norms. The authors leave unresolved whether the representation can be chosen so that this summable ideal is non-trivial.