Minimum continuum compatible with vanishing of all higher derived limits of A

Determine the least possible value of 2^{\aleph_0} consistent with ZFC together with the assertion that, for every integer n with 1 \leq n < \omega, the derived limit \lim^n A equals 0, where A denotes the inverse system of abelian groups indexed by ({^{\omega}\omega}, \leq) formed from the sets I_f = {(k,m) \in \omega \times \omega \mid m < f(k)} with groups A_f = \bigoplus_{I_f} \mathbb{Z} and restriction bonding maps.

Background

The paper recalls two motivating questions about the vanishing and nonvanishing of higher derived limits of the inverse system A. The first half of Question 1 asks for the minimum value of the continuum compatible with \"for all 1 \leq n < \omega, \lim^n A = 0\".

The authors resolve the second half of Question 1 (the version for all abelian groups H) by proving a lower bound 2^{\aleph_0} \geq \aleph_{\omega+1}, but they explicitly note that the first half concerning A (with H = \mathbb{Z}) remains open.