Consistency of simultaneous vanishing of lim^n A_kappa for all cardinals

Determine whether it is consistent with ZFC that, for every cardinal κ and every integer n with 1 ≤ n < ω, the n-th right derived inverse limit lim^n of the inverse system A_κ (the _κ system with (A_κ)_{α,k} = ℤ^k and canonical projection maps indexed over ω^κ) vanishes, i.e., lim^n A_κ = 0; equivalently, ascertain whether in such a model strong homology is additive and has compact supports on the class of locally compact metric spaces.

Background

The paper proves that, for each fixed cardinal κ, the additivity and compact supports of strong homology on locally compact metric spaces of weight at most κ are equivalent to the vanishing of all higher derived limits of the inverse system A_κ. This closes a circle connecting homological additivity with set-theoretic properties of derived limits.

Known results provide, for any given κ, forcing extensions where lim^n A_κ = 0 for all 1 ≤ n < ω, though these extensions typically add many reals. The open problem asks whether one can achieve the vanishing simultaneously for all κ within a single model of ZFC, which would yield a global additivity and compact supports property for strong homology across all locally compact metric spaces.