Implication from vanishing of lim^n A to vanishing of lim^n A[H] for all abelian groups

Determine whether the global vanishing lim^n A = 0 for all integers n with 1 ≤ n < ω implies lim^n A[H] = 0 for all 1 ≤ n < ω and for every abelian group H, where A[H] denotes the inverse system obtained by replacing the basic groups in A by copies involving H.

Background

In known models where lim^n A vanishes simultaneously for all n, the same has been shown to hold for A[H] for any abelian group H. The authors ask whether this implication holds in general, i.e., whether the vanishing of derived limits for A alone forces vanishing for all A[H] without additional assumptions.

A positive resolution would unify the behavior of derived limits across a broad class of related systems and sharpen known bounds on the continuum compatible with simultaneous vanishing.