Positive Rates Conjecture for simple interacting particle systems

Let an IPS be simple when it is one-dimensional, homogeneous, and has one-sided nearest-neighbor interactions, meaning $\Lambda=\mathbb{Z}$, $N_j=\left\{j,\,j+1\right\}$, and $P_j\left(\,\cdot\mid\zeta\right)=P_0\left(\,\cdot\mid T_{-j}(\zeta)\right)$. Its transition rates are positive when the transition matrix differs from the identity on every entry, and it is ergodic when it has an attractive distribution.

PRC for simple IPS. Every simple IPS with positive rates is ergodic.

This is the basic two-state, one-sided nearest-neighbor case of the Positive Rates Conjecture. The paper identifies simple IPS behaving like noisy versions of the East model as the remaining nontrivial cases, but the supplied text does not establish whether this restricted conjecture has been resolved.