Local B-ideal structure of square sums

Determine whether, for every square sum I of two B-ideals and every I-positive set Y, there is an I-positive subset X of Y such that the restriction I↾X is a B-ideal.

Background

Proposition 5.31 establishes an analogous local non-pathology result for Q-coloring ideals. The authors formulate the corresponding question for arbitrary square sums of B-ideals.

References

Question 5.32. Let I be the square sum of two B-ideals and Y ∉ I. Is there X ⊆ Y with X ∉ I such that I ↾ X is a B-ideal (equivalently, non-pathological)?

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 5.32, p. 31