Characterizing E_{I,ε} as S_{I,ε} ∩ E

Determine whether, for every S^⋆-contributive pair (I, ε) with I a partition of ω into finite nonempty intervals and ε ∈ ℓ^1_+, the equality E_{I,ε} = S_{I,ε} ∩ E holds, where E denotes the σ-ideal generated by closed null subsets of 2^ω.

Background

The paper shows that E_{I,ε} ⊆ S_{I,ε} ∩ E, and that both families refine the classical σ-ideal E of Fσ-measure-zero sets. Whether the reverse inclusion always holds is left unresolved.

Establishing or refuting E_{I,ε} = S_{I,ε} ∩ E would clarify the precise position of E_{I,ε} within the intersection of smallness notions and sharpen the structural comparison between the two layered ideals.