Censoring symmetry conjecture for Glauber dynamics on transitive graphs

Let $\{G_n\}$ be a sequence of transitive graphs, and consider the Glauber dynamics for the Ising model on these graphs. Let $\beta_c$ denote the critical temperature, let $\delta$ satisfy $|\delta|<\beta_c$, and set

$$\beta_1=\beta_c-\delta,\qquad \beta_2=\beta_c+\delta.$$

A suitable notion of censoring is assumed for the censored dynamics. Censoring symmetry conjecture. There is cutoff for the original dynamics at $\beta_1$ if and only if there is cutoff for the censored dynamics at $\beta_2$.

This conjecture expresses the expected symmetry of Glauber dynamics around the critical temperature, extending the behavior observed for the mean-field Ising model to transitive graph sequences. The appropriate notion of censoring and the general validity of the equivalence remain open.