B-equivalence of sequences

Determine the descriptive and structural complexity of the equivalence relation on Banach-space sequences defined by x∼B y if and only if B(x)=B(y).

Background

Different sequences, and even sequences in different Banach spaces, may represent the same B-ideal. The paper explicitly asks how complex the resulting equivalence relation is.

References

Question 2.7. Let us say that two sequences x = (xn) and y = (yn) in a Banach space X are B-equivalent if B(x) = B(y). How complex is this equivalence relation?

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 2.7, Section 2, page 7