Square sums of B-ideals

Determine whether the square sum of two B-ideals is necessarily a B-ideal.

Background

For comparability colorings, the associated coloring ideal is expressed as the square union of two B-ideals. The authors establish this representation but do not know whether square sums preserve the class of B-ideals.

References

Question 5.27. Is the square sum of B-ideals an B-ideal?

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 5.27, Section 5.2, page 30

We do not know the answer of the following.Question 5.27. Is the square sum of B-ideals an B-ideal?

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 5.27, Section 5.2, page 30