Semiproperness of the forcing P(A) that adds a closed copy of ω1 inside A⊆E^κ_ω

Determine whether, for every regular cardinal κ ≥ ω2 and every stationary subset A ⊆ E^κ_ω, the forcing P(A) consisting of normal functions p: α+1 → A for some α < ω1 (ordered by end-extension) is semiproper. Establish whether P(A) is semiproper in full generality or identify precise conditions on κ and A under which semiproperness holds or fails.

Background

In Section 2 the paper studies the natural forcing P(A) which, given a stationary set A ⊆ E^κ_ω for a regular κ ≥ ω2, adds a closed subset of A of order type ω1 by conditions that are normal functions p: α+1 → A ordered by end-extension.

The authors note that in most cases P(A) collapses κ to ω1 and is not proper (whenever E^κ_ω \ A is stationary). While they establish that P(A) preserves stationary subsets of ω1, they explicitly state that it is unknown whether P(A) is always semiproper. This uncertainty persists even though P(A) satisfies Shelah’s S-condition, which guarantees preservation of ω1 and not adding reals but does not by itself imply semiproperness.

Clarifying the semiproperness status of P(A) would refine our understanding of iterations involving P(A), their preservation properties, and their compatibility with forcing axioms tied to semiproperness.