Borel complexity of summable representations

Determine whether the set of sequences g in (C(2^N))^N for which B(g) is summable is Borel.

Background

The paper provides explicit representations of non-pathological Fσ ideals in C(2N). This raises a descriptive-set-theoretic question about the complexity of the subfamily consisting of summable B-ideals.

References

Question 4.7. Is {g = (g_n)_n ∈ (C(2N))N ∶ B(g) is summable} a Borel set?

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 4.7, Section 4.1, page 17