Sub-exponential perturbation-rate conjecture for TASEP-H

Let $\rho$ be the particle density, let $h$ be the critical density, let $\phi(\rho;n)$ be the finite-$n$ typical flux, and let $\phi^*$ be the maximal-equilibrium-state flux. Let $g$ be a positive non-increasing function with $g(n)\to0$ and

$$\lim_{n\to\infty}\frac{\log g(n)}{n}=0.$$

Thus $g(n)$ is sub-exponential. Consider the model with either A-perturbations or I-perturbations, with perturbation probability $\rho n g(n)$ at a given time under the corresponding perturbation rule. Sub-exponential perturbation-rate conjecture. The limiting typical flux satisfies

$$\phi(\rho)=\lim_{n\to\infty}\phi(\rho;n)=\rho$$

for $0<\rho<h$, and

$$\phi(\rho)=\lim_{n\to\infty}\phi(\rho;n)=\phi^*$$

for $h<\rho<1/2$. The conjecture predicts that sub-exponential perturbation rates preserve the low- and high-density flux regimes established for the basic perturbation mechanisms.