Global separation between invariants of N and E_{I,ε} for all parameter pairs

Determine whether there exists a model of ZFC in which cov(N) < cov(E_{I,ε}) holds for all pairs (I, ε) with I a partition of ω into finite nonempty intervals and ε ∈ ℓ^1_+, and dually whether there exists a model in which non(E_{I,ε}) < non(N) holds for all such pairs.

Background

The σ-ideal E_{I,ε} is defined via interval partitions I of ω and summable sequences ε, capturing sets X ⊆ 2^ω that are eventually included coordinatewise in suitably small families of basic clopen pieces. The classic null ideal N serves as a benchmark in Cichoń’s diagram. The paper establishes ZFC comparabilities involving these families and presents forcing models locating their characteristics relative to those of N.

This question asks for robust “global” consistency separations between cov(N) and cov(E_{I,ε}) (and dually, between non(E_{I,ε}) and non(N)) that hold uniformly for all E-contributive parameter pairs (I, ε). It seeks to understand whether the new ideals can be consistently placed strictly above or below N across all instances.