Mixing time for the asymmetric simple exclusion process in a random environment

We consider the simple exclusion process in the integer segment $[1,N]$ with $k\le N\mathrm{/}2$ particles and spatially inhomogenous jumping rates. A particle at site $x\in [1,N]$ jumps to site $x-1$ (if $x\ge 2$) at rate $1-{\omega }_x$ and to site $x+1$ (if $x\le N-1$) at rate ${\omega }_x$ if the target site is not occupied. The sequence $\omega =({\omega }_x{)}_{x\in \mathbb{Z}}$ is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume $\mathbb{E}[\log {\rho }_1]<0$ where $\rho_1:= (1-\omega_1)/\omega_1$, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of $N$. More precisely, for the exclusion process with $N^{\beta +o(1)}$ particles where $\beta \in [0,1)$, we have in the large $N$ asymptotic $$N^{\max (1,\frac{1}{\lambda },\beta +\frac{1}{2\lambda })+o(1)}\le t_{\mathrm{M}\mathrm{i}\mathrm{x}}^{N,k}\le N^{C+o(1)}$$ where $\lambda >0$ is such that $\mathbb{E}[{\rho }_1^{\lambda }]=1$ ($\lambda =\mathrm{\infty }$ if the equation has no positive root) and $C$ is a constant which depends on the distribution of $\omega$. We conjecture that our lower bound is sharp up to sub-polynomial correction.