Wagner's Rayleigh-to-ULC Big Conjecture

Let $\mathfrak{P}_n$ be the probability measures on subsets of $[n]$, and let a Rayleigh measure be a measure satisfying the Rayleigh negative-dependence inequalities. Let ULC mean that its rank sequence is ultra-log-concave. Wagner's Big Conjecture. Every Rayleigh measure $\mu\in\mathfrak{P}_n$ is ULC. The source later gives counterexamples to this conjecture, so it is refuted.