Pemantle's CNA+-to-ULC conjecture

Let $\mathfrak{P}_n$ be the probability measures on subsets of $[n]$, and let CNA+ denote the strongest negative-association property used in the source. Let ULC mean that the rank sequence is ultra-log-concave. CNA+-to-ULC conjecture. If $\mu\in\mathfrak{P}_n$ is CNA+, then it is ULC. The source constructs examples showing that CNA+ need not even imply the weaker SLC property, so this conjecture is refuted.