Separation results for σ-closed and ccc posets analogous to Corollary 3.12

Ascertain whether there exist theorems that yield non-implications analogous to those established in Corollary 3.12 for P equal to the class of all σ-closed posets and for P equal to the class of all ccc posets. Concretely, determine whether one can prove consistency results separating the restricted Maximality Principle MP*(P,H(κrefl),Γ) from the Recurrence Axioms (P,∅)-Π2-RcA and (P,∅)-Σ2-RcA in these two settings, as was shown for proper, semiproper, and stationary preserving posets.

Background

In Section 3, the authors develop hierarchies of restricted Recurrence Axioms and Maximality Principles and prove separations between them via compatibility with the Ground Axiom. Corollary 3.12 establishes strong non-implications for classes P such as proper, semiproper, and stationary preserving posets.

The stated problem asks whether similar separation results can be obtained when P is instead the class of all σ-closed posets or the class of all ccc posets, extending the scope of the separation phenomena beyond the cases handled in Corollary 3.12.