Preservation under the Laver property: keeping cov(E_{I,ε}) small

Investigate whether forcing notions satisfying the Laver property necessarily preserve small values of the covering number cov(E_{I,ε}) for all pairs (I, ε) with I a partition of ω into finite nonempty intervals and ε ∈ ℓ^1_+.

Background

It is known that forcing notions with the Laver property do not increase the covering of the σ-ideal E (generated by closed null sets). Given the parallels established in the paper between E and the layered ideals E_{I,ε}, it is natural to ask whether the same preservation holds for E_{I,ε}.

A positive answer would yield robust preservation theorems for the covering numbers of these refined ideals under a broad class of proper, bounding forcings.