Consistency of nonvanishing for all higher derived limits including k = 1

Determine whether it is consistent with ZFC that \lim^k A \neq 0 holds for every integer k with 1 \leq k < \omega, 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

Question 2 asks about realizing arbitrary vanishing patterns and, in particular, whether one can have \lim^n A \neq 0 for all n. The authors construct models where \lim^k A \neq 0 for all k \geq 2 but also have \lim^1 A = 0.

They explicitly state that the \"in particular\" clause—nonvanishing for all k including 1—remains open.