Strictness of the Σ_n-correct forcing axiom hierarchy for n>2

Determine whether, for n>2 and any n‑nice forcing class Γ, the axiom Σ_{n+1}–CFA_{<κ}(Γ) is strictly stronger than Σ_n–CFA_{<κ}(Γ), i.e., whether Σ_{n+1}–CFA_{<κ}(Γ) does not follow from Σ_n–CFA_{<κ}(Γ).

Background

The chapter examines partial separations showing that Σn–CFA{<κ}(Γ) need not imply Σ{n+2}–CFA{<κ}(Γ). The remaining open issue is whether consecutive levels are strictly separated for n>2.