Borel complexity of summable B-representations

Determine whether the set of sequences g in C(2^N) whose B-ideal B(g) is summable is Borel.

Background

The paper gives explicit representations of non-pathological Fσ ideals in C(2N). This raises a descriptive-set-theoretic question about the complexity of the subclass consisting of sequences whose represented ideal is summable.

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, p. 17