Complexity of B-equivalence

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

Background

The paper observes that B-representations are highly non-unique and defines B-equivalence for two representing sequences that generate the same B-ideal. The complexity of this equivalence relation is left unresolved.

References

Question 2.7. Let us say that two sequences x = (x_n) and y = (y_n) 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, p. 7