Negative association under symmetric exclusion evolution

Let a symmetric exclusion process start from a deterministic initial configuration. Write $\eta_t$ for its configuration at time $t$, and say that a distribution is negatively associated if increasing functions depending on disjoint sets of coordinates have nonpositive covariance. Negative-association conjecture. For every $t>0$, the distribution of $\eta_t$ is negatively associated. The conjecture concerns a fundamental negative-dependence property for interacting particle systems; the source states that a stronger form is proved in the paper.