Minimum continuum size compatible with simultaneous vanishing of lim^n A

Ascertain the least possible value of the continuum 2^{ℵ_0} that is compatible with the assertion that lim^n A = 0 for every integer n with 1 ≤ n < ω.

Background

The authors identify this as a major open question. Current results provide an upper bound of ℵ_{ω+1} for the continuum in models where all higher derived limits of A vanish. Establishing the optimal lower bound would clarify the exact set-theoretic strength required for simultaneous vanishing.

A positive answer to the implication question about A[H] would render ℵ_{ω+1} optimal, tying the continuum's size tightly to the behavior of derived limits in these systems.