Higher limits of wider systems

Write ${\mathbf{A}}_{\lambda }$ for what might be described as the most elementary nontrivial inverse system of abelian groups indexed by the functions from the cardinal $\lambda$ to the set of natural numbers. The question of whether for any fixed $n$ the derived limit ${\mathrm{l}\mathrm{i}\mathrm{m}}^n{\mathbf{A}}_{\lambda }$ may vanish for only a nonempty subset of the class of infinite cardinals $\lambda$ is recorded in both [Be17] and [Ban23], and bears closely on several related further ones. We answer this question in the affirmative; in fact, we show the maximal possibility, namely that this can simultaneously happen in every degree $n>1$.