Defectlessness from adapted top-degree-p slices

Identify additional hypotheses under which defectlessness of all adapted top degree-$p$ extensions $L/K_{H_i}$ implies defectlessness of the full finite Galois extension $L/K$.

Background

For a ramification-adapted decomposition of an elementary abelian Galois group, the paper proves that all ramification ideals are principal exactly when every associated degree-$p$ extension $L/K_{H_i}$ is defectless. Kuhlmann’s degree-$p^2$ example shows that these top slices can all be defectless while the full extension has defect. The unresolved issue is to find supplementary assumptions that eliminate this phenomenon.