An upper bound conjecture for mixing times in ballistic random environments

Let $N$ denote the size of the segment, and consider the simple exclusion process in a ballistic random environment. Its mixing time is the time needed for the process to approach its stationary distribution. Upper bound conjecture. For any ballistic random environment, the mixing time of the respective exclusion process is with high probability at most of order $N^{\frac{3}{2}}$. The paper gives the correct order of the mixing time only in the non-nestling case; this conjectural upper bound concerns the general ballistic regime, where the correct order remains open.