Mixing time under monotone censoring

We prove that the lazy random walk on the discrete cube, censored to any increasing set of fixed positive density, mixes in time $O(n\log n)$, answering a question of Ding and Mossel. More precisely, for every nonempty increasing set $A\subseteq \{0,1\}^n$, \begin{equation} t_{\mathrm{mix}}(P) \le K