ULC+ implies negative association

Let $\mu$ be a measure on a Boolean lattice. It is ULC+ when every measure obtained from it by imposing an external field and projecting is ULC. ULC implication conjecture. If $\mu$ is ULC+, then $\mu$ is CNA, hence CNA+, and in particular $\mu$ is negatively associated. The paper describes this as philosophically useful but difficult to apply because verifying ULC+ is hard; it was presented as open.