Equivalence of CNA+, JNRD+ and h-NLC+

The properties CNA+, JNRD+ and h-NLC+ are stronger versions of CNA, JNRD and hereditary NLC, respectively, requiring the corresponding property after every imposition of an external field and projection. Equivalence conjecture. All three properties CNA+, JNRD+ and h-NLC+ are equivalent. The horizontal implications among these properties were open in the paper; the claim concerns whether the three strengthened notions define the same class of measures.