Pemantle's conjecture that CNC implies CNA

Let $cmu a$ be a probability measure on $c{0,1}^n$. It is CNC if every pair of disjoint increasing events is negatively correlated, and it is CNA if every pair of disjoint subsets of coordinates has negatively associated restrictions under conditioning on any increasing function of the remaining coordinates. In the setting of generalized urn measures, write $R^+$ for the property that every external field transformation $W\circ\mu$, with $W_i\in\{0\}\cup[1,\infty)$ for all $i$, is negatively correlated.

Pemantle's conjecture. CNC implies CNA. In particular, the equivalent formulation considered here is that $R^+$ implies CNA.

The conjecture asks whether the stronger conditional negative association property follows from conditional negative correlation. The paper notes that a negative answer for generalized urn measures would provide a counterexample to the CNC-implies-CNA conjecture, and that CNC for this class is equivalent to the stated $R^+$ property; its resolution is not supplied here.