Sigma-Prikry forcing II: Iteration Scheme

In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class notions which call [Formula: see text]-Prikry, showed that many the known Prikry-type center around singular cardinals countable cofinality are text]-Prikry. We given text]-Prikry poset text] text]-name for non-reflecting stationary set text], there exists corresponding projects to kills stationarity text]. paper, develop general scheme iterating posets and, as an application, blow up power limit Laver-indestructible supercompact cardinals, then iteratively kill all subsets its successor. This yields model in cardinal hypothesis fails simultaneous reflection finite families sets holds.