Nontrivial summable extension under cotype

Determine whether a B-representation in a Banach space with non-trivial cotype can be chosen so that the associated summable ideal fSum((||x_n||^q)_n) is non-trivial.

Background

For a B-ideal represented by a sequence in a Banach space with non-trivial cotype q, the paper proves the inclusion B((x_n)_n) c Sum((||x_n||q)_n). The unresolved issue is whether the representation can be selected so that the containing summable ideal is non-trivial rather than equal to the whole power set or otherwise vacuous.

References

However, it remains an open question whether one can choose the B-representation such that this summable ideal is non-trivial.

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Section 1, page 3; reiterated as Question 3.9, Section 3.1, page 12