Hereditarily pathological Fσ ideals

Construct or determine the existence of an Fσ ideal such that every restriction to a positive set is pathological.

Background

The paper introduces the dual notions of local and hereditary non-pathology. It asks whether the opposite extreme exists: an Fσ ideal whose every positive restriction remains pathological.

References

Question 5.33. Does there exist a hereditarily pathological Fσ-ideal? Specifically, is there an Fσ-ideal such that every restriction to a positive set is pathological?

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 5.33, Section 5.2, page 31