Sub-exponential slowing conjecture for the zero-range model

Let $\rho$ be the particle density, let $h$ be the critical density, let $\phi_{\pi(n)}(\rho;n)$ be the finite-$n$ flux in the zero-range model with parameter $\pi(n)$, and let $\phi^*$ be the maximal-equilibrium-state flux. Suppose that

$$1-\pi(n)=\operatorname{SUBEXP}(n),$$

meaning that the quantity on the right is sub-exponential in the sense specified in the source. Sub-exponential slowing conjecture. For the zero-range model,

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

when $0<\rho<h$, while

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

when $h<\rho<1/2$. This conjecture extends the proposed perturbation-rate behavior to the zero-range formulation, with the parameter approaching $1$ sub-exponentially.