Intermediate behavior of forceably Σ_{n−1}-correct regular cardinals under Σ_n–CFA

Determine whether there exists a model of the Σ_n-correct forcing axiom Σ_n–CFA_{<κ}(Γ) (for n>2 and any n‑nice forcing class Γ) in which the class of regular cardinals that are forceably Σ_{n−1}-correct, i.e., ♦C^{(n−1)}∩Reg, is neither empty nor a proper class.

Background

Section 3.3 studies whether increasing n in Σ_n-correct forcing axioms yields strictly stronger principles. The author introduces the class ♦C^{(m)} of cardinals that become Σ_m-correct in some forcing extension.

Establishing a model where ♦C^{(n−1)}∩Reg is neither empty nor a proper class would clarify whether the hierarchy can exhibit intermediate patterns at level n>2 and would inform strictness results between successive levels.