Cotype representation with nonsummable q-th norms

Determine whether a B-ideal represented in a Banach space of non-trivial cotype q can be represented by a sequence (x_n)_n for which (||x_n||^q)_n is not summable.

Background

The paper proves the inclusion B(x)⊆Sum((||x_n||q)_n) for representations in spaces of non-trivial cotype q. Question 3.9 asks whether the q-power norm sequence can nevertheless fail to be summable for a suitable representation.

References

Question 3.9. Suppose that an ideal is B-representable on a space with non trivial cotype q. Is it possible to do it in such a way that the corresponding sequence (xn)n satisfies that (YxnYq)n is not summable?

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 3.9, Section 3.1, page 12