Strong tree property and the narrow system property

Establish whether the strong tree property implies the narrow system property.

Background

The narrow system property is a weakening of the strong tree property that holds in all known models of the strong tree property and is useful in deriving SCH and related consequences. The paper notes that it is generally easier to obtain than the strong tree property and can hold in models where the strong tree property fails.

The implication from the strong tree property to the narrow system property remains unresolved. Proving it would yield a positive answer to the paper’s question concerning SCH above a cardinal with the strong tree property.

References

It is open if the strong tree property implies this narrow system property; if it does, then Lambie-Hanson's result would answer Question \ref{q:doesTPimplySCH} immediately.

— A Note on Narrow Systems  (2610.00934 - Adkisson, 1 Oct 2026) in Section 1, Introduction

This motivates the following question: Assuming the strong tree property, do all $(,)$-trees generated from narrow concrete $(,)$-systems have a cofinal branch?

— A Note on Narrow Systems  (2610.00934 - Adkisson, 1 Oct 2026) in Section 1, Introduction

Does the strong tree property at $$ imply the hypotheses of Theorem \ref{thm:LH}?

— A Note on Narrow Systems  (2610.00934 - Adkisson, 1 Oct 2026) in Section 2, subsection “Two-cardinal objects,” immediately after Theorem \ref{thm:LH}