Reversible-scale conjecture for symmetric condensate transitions

Let $S_{\star}$ be the set of sites on which the condensate may reside, and let $r(x,y)$ be the jump rates of the underlying random walk. Let $Y^{\mathrm{rv}}(\cdot)$ be the Markov chain on $S_{\star}$ with rates $a^{\mathrm{rv}}(x,y)=r(x,y)$, and set $\theta_N^{\mathrm{rv}}=1/d_N$. Reversible-scale conjecture. Suppose that

$$r(x,y)=r(y,x)\quad\text{for all }x,y\in S_{\star}$$

and $\lim d_N\log N=0$. Then the movement of the condensate is described by the Markov chain $Y^{\mathrm{rv}}(\cdot)$ on $S_{\star}$ with scale $\theta_N^{\mathrm{rv}}$. When the antisymmetric rates vanish on $S_{\star}$, the non-reversible scale is too short to observe transitions, and the reversible scale is expected to be the correct one; this remains unproved in the source.