Lacoin's mixing-time conjecture for asymmetric exclusion on the circle

Let $N$ be the size of the circle and let $k$ be the number of particles in an asymmetric simple exclusion process, with particles moving clockwise at rate $p$ and counterclockwise at rate $1-p$ for some $p\in(1/2,1]$. The particles obey the exclusion constraint.

Lacoin's conjecture. The mixing time is of order

$$N^{2}\min(k,N-k)^{-1/2}.$$

Moreover, the cutoff phenomenon does not occur.

This conjecture concerns the expected mixing behavior of asymmetric exclusion processes on a circle and is attributed to Hubert Lacoin. Its resolution is not stated in the source.