Consistency of MP* Π2 without MP* Σ2 at H(κrefl)

Determine whether it is consistent that the persistent Maximality Principle MP*(P,H(κrefl),Π2) holds while MP*(P,H(κrefl),Σ2) fails for some iterable class P of posets.

Background

Section 6 develops another hierarchy of restricted Maximality Principles based on provably P-persistent formulas (MP*). The authors prove multiple separations among levels and types but note remaining gaps.

This problem targets a specific separation at level 2 of the hierarchy, asking whether a model can satisfy MP* for Π2 formulas over H(κrefl) yet fail MP* for Σ2 formulas, thereby sharpening understanding of the relative strength of these restricted principles.