Consistency separation between covering and uniformity for E_{I,ε} (and analogously for S_{I,ε})

Determine whether there exists a model of ZFC in which, for some or for all pairs (I, ε) with I a partition of ω into finite nonempty intervals and ε ∈ ℓ^1_+, the covering number cov(E_{I,ε}) is strictly less than the uniformity number non(E_{I,ε}); and, dually, whether there exists a model in which non(E_{I,ε}) < cov(E_{I,ε}). Formulate and resolve the analogous consistency questions for the σ-ideals S_{I,ε}.

Background

The paper introduces two σ-ideals on 2^ω parametrized by a partition I of ω into finite intervals and a summable positive sequence ε: S_{I,ε} and E_{I,ε}. These ideals refine the classical ideals of small sets and Fσ-measure-zero sets, and the authors analyze their associated cardinal characteristics (additivity, covering, uniformity, and cofinality) and their relations to localization/anti-localization cardinals and to Cichoń’s diagram.

The question asks about separating the two middle characteristics (covering and uniformity) for the same ideal, in the spirit of separations familiar from Cichoń’s diagram. It seeks models where cov(E_{I,ε}) < non(E_{I,ε}), and also models with the dual inequality non(E_{I,ε}) < cov(E_{I,ε}). The same problem is posed for S_{I,ε}. These problems probe whether the newly introduced ideals admit the same fine-grained calibrations of cardinal characteristics as the classical null and meager ideals.