Complexity of non-pathological Fσ ideals

Determine the Borel or projective complexity of the collection of non-pathological Fσ ideals, specifically the set {A∈K(2^N):⟨A⟩ is non-pathological}.

Background

Every Fσ ideal can be generated by a closed hereditary collection of subsets of N. Since the paper constructs both pathological and non-pathological examples, it asks for the descriptive complexity of the class of generators producing non-pathological ideals.

References

Question 5.20. What is the (Borel or projective) complexity of the collection of non-pathological Fσ-ideal? More precisely, what is the complexity of the following collection: {A ∈ K(2N) : ⟨A⟩ is non-pathological} where ⟨A⟩ denotes the ideal generated by A (recall, that every Fσ ideal can be generated by a closed collection of subsets of N).

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 5.20, p. 28