Representation of tall B-ideals in c0-saturated spaces

Prove or disprove that every tall B-ideal admits a B-representation in a Banach space saturated by isomorphic copies of c0.

Background

Theorem 3.2 characterizes tall B-ideals through sequences whose subsequences contain further subsequences equivalent to the unit basis of c0. The paper asks whether this sequential saturation can be strengthened to saturation of the whole ambient space by copies of c0.

References

We do not know if this can be strengthened to assert that tall B-ideals can be B-represented in spaces that are saturated by isomorphic copies of c0, i.e. such that every infinite-dimensional subspace has a subspace isomorphic to c0.

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Section 1, page 3