Non-reversible condensate movement conjecture

Let $S$ be the finite state space of the underlying random walk, with jump rates $r(x,y)$, and let $S_{\star}$ be the set of sites on which the condensate may reside. Define the Markov chain $Y^{\mathrm{nrv}}(\cdot)$ on $S_{\star}$ by the rates

$$a^{\mathrm{nrv}}(x,y)=\left[r(x,y)-r(y,x)\right]\mathbf{1}\left\{r(x,y)>r(y,x)\right\},$$

and set $\theta_N^{\mathrm{nrv}}=1/(Nd_N)$. Here $d_N$ is the inclusion-process parameter and $N$ is the number of particles. Non-reversible condensate movement conjecture. Suppose that $\lim d_N\log N=0$. Then the movement of the condensate is described by the Markov chain $Y^{\mathrm{nrv}}(\cdot)$ with scale $\theta_N^{\mathrm{nrv}}$. This predicts that non-reversible condensate transitions occur on a scale $O(N)$ faster than in the reversible case; the paper presents heuristic support but no proof in the general setting.