Separation between invariants of N and S_{I,ε}

Determine whether there exists a model of ZFC in which cov(N) < cov(S_{I,ε}) for some or for all pairs (I, ε) with I a partition of ω into finite nonempty intervals and ε ∈ ℓ^1_+, and analogously investigate the dual separation at the uniformity coordinate.

Background

The σ-ideal S_{I,ε} is the ‘antilocalization’ counterpart to E_{I,ε}, capturing sets X ⊆ 2^ω that meet small families of clopen pieces infinitely often. The paper situates S_{I,ε} within and relative to N and E and studies its cardinal characteristics.

This question mirrors the prior ones for E_{I,ε}, now asking whether cov(S_{I,ε}) can be strictly above cov(N) (and dually at the uniformity coordinate), either for some specific parameter pairs or uniformly across all pairs.