Crossover-regime mixing-time conjecture for the weakly asymmetric simple exclusion process

Let $b_N$ and $k_N$ satisfy

$$\lim_{N\to\infty}\frac{b_NN}{\log k_N}=\beta$$

for some $\beta\in(0,\infty)$, with $\lim_{N\to\infty}k_N/N=0$. Write $T_{\rm mix}^{N,k_N}(\varepsilon)$ for the mixing time of the weakly asymmetric simple exclusion process. Crossover-regime mixing-time conjecture. For every $\varepsilon>0$,

$$\lim_{N\to\infty}\frac{T_{\rm mix}^{N,k_N}(\varepsilon)\log k_N}{N^2}=\begin{cases}\frac{2}{\beta}+\frac{1}{\beta^2},&\text{if }\beta\;\leqslant\; 1/2,\left(\frac{\sqrt{2}+2\sqrt{\beta}}{2\beta}\right)^2,&\text{if }\beta\;\geqslant\; 1/2. \end{cases}$$

This gives a conjectural mixing-time asymptotic in the crossover regime $b_N\asymp\log k_N/N$; the paper provides heuristic support, while the assertion remains unproved in the stated generality.