The equivalence of CNC and CNA and of NC+ and NA+

Let $\mu$ be a measure on a finite Boolean cube. The properties CNC, CNA, NC+, and NA+ are the correlation properties defined in the paper, with “+” denoting closure under external fields and conditioning. Equivalence conjecture. (a) The properties CNC and CNA are equivalent. (b) The properties NC+ and NA+ are equivalent. This would extend the equivalence known for symmetric measures; the claim is stated as an interesting general conjecture, with no resolution given here.