Optimal-mixing conjecture for Kawasaki dynamics above the analytic threshold

Let $\mathcal G_\Delta$ be the class of graphs under consideration, let $\hat{\mu}_{G,\beta,\eta}$ be the fixed-magnetization Ising measure on $G\in\mathcal G_\Delta$, and let $\beta_u$ and $\eta_a$ denote the uniqueness and analytic thresholds, respectively. Optimal-mixing conjecture. If $0\leq\beta<\beta_u$ or if $\beta>\beta_u$ and $|\eta|>\eta_a$, then the Kawasaki dynamics for $\hat{\mu}_{G,\beta,\eta}$ are optimally mixing: for every $G\in\mathcal G_\Delta$, the mixing time is in $O(|V(G)|\cdot\log(|V(G)|))$. This is presented as an improvement of the paper's rapid-mixing theorem; the supplied text gives no resolution, so the conjecture remains open.