A structural monotonicity criterion ensuring inclusion between layered ideals

Identify a natural relation R on partitions I, J of ω into finite nonempty intervals such that, whenever I R J, the inclusion S_{I,ε} ⊆ S_{J,ε} holds for all ε ∈ ℓ^1_+, and analogously find such a criterion for E_{I,ε} ⊆ E_{J,ε}.

Background

Section 3 analyzes when S_{I,ε} ⊆ S_{J,ε} or E_{I,ε} ⊆ E_{J,ε} should hold, and shows that several natural relations (like refinement or the preorder ⊑) are insufficient by themselves. The authors provide sufficient conditions that also involve ε.

This question seeks an intrinsic and ‘reasonable’ monotone relation on partitions alone that would imply inclusion of the corresponding ideals uniformly in ε, clarifying the structural dependence of the ideals on the underlying interval partition.