Stationarity of internally club sets under ν⁺-closed forcing

Determine whether, for any ν⁺-closed forcing poset P and any stationary set S ⊆ P_ν(H(Θ)) consisting of subsets N of H(Θ) of cardinality less than ν that are internally club (i.e., [N]^{<ν} ∩ N contains a club in [N]^{<ν}), the set S remains stationary in the forcing extension V^P.

Background

The paper develops a new variant of Mitchell forcing to separate internally club from internally approachable across an infinite interval of cardinals and analyzes preservation under various forcing notions. The authors introduce a strengthened form ICNIA⁺ and show downward preservation under sufficiently distributive forcings.

The posed question concerns preservation of stationarity for sets of internally club subsets of H(Θ) within P_ν(H(Θ)) under ν⁺-closed forcing. It asks whether strong closure alone guarantees that such a stationary set remains stationary after forcing, connecting to broader themes of stationarity preservation in Jech’s framework for P_ν(X).