Complexity comparison on the Cantor space: ideal vs L(I)

Establish whether, for the Cantor space X=2^ω and any ideal I on ω, the inclusion I ⊆ L_X(I) holds when both are viewed as subsets of 2^ω; and, in a weaker form, determine whether there exists any ideal I such that the family of sets of I-cluster points satisfies 6_X(I)=P(X).

Background

The paper discusses a heuristic that L(I) may have lower complexity than I. This motivates asking for a direct comparison, especially in the canonical Polish space 2^ω, between the ideal I (seen as a subset of 2^ω via characteristic functions) and the family L_X(I).

The authors also pose a weaker existence problem: whether any ideal can have its cluster-point family 6_X(I) equal the entire power set of X, which would be an extreme case of expressivity for cluster points.