Coherent sequences and threads

The combinatorial principle □(λ) says that there is a coherent sequence of length λ that cannot be threaded. If λ=κ+, then the related principle □κ implies □(λ). Let κ⩾ℵ2 and X⊆κ. Assume both □(κ) and □κ fail. Then there is an inner model N with a proper class of strong cardinals such that X∈N. If, in addition, κ⩾ℵ02 and n<ω, then there is an inner model Mn(X) with n Woodin cardinals such that X∈Mn(X). In particular, by Martin and Steel, Projective Determinacy holds. As a corollary to this and results of Todorcevic and Velickovic, the Proper Forcing Axiom for posets of cardinality +(ℵ02) implies Projective Determinacy.