The ULC conjecture for negatively associated measures

For a measure on a Boolean lattice, its rank sequence is the sequence of probabilities of the possible values of the total number of occupied coordinates. It is ULC (ultra-log-concave) when the nonzero terms of the rank sequence, after division by the corresponding binomial coefficients, form a log-concave sequence with interval support. ULC conjecture. The strongest version asserts that every negatively associated measure is ULC. Weaker versions replace negative association by any of the other six stronger conditions in Figure 1, namely CNA+, JNRD+, h-NLC+, CNA, JNRD or h-NLC. The paper states that these conjectures had been checked only for lattices of rank at most $4$, so their general validity remained open.