Universal Airy-one particle-position scaling conjecture for exclusion processes

Let $\eta_t$ be a Markov process with generator specified by the paper, with particle positions satisfying $X_n(0)=-\lfloor n/\rho\rfloor$ for $\rho\in(0,1)$. Let $\mu_\rho$ be an invariant measure satisfying the paper's Assumption~, and let $J(\rho)$, $\Gamma(\rho)$ and $A(\rho)$ be the associated current and KPZ scaling parameters. Define

$$\theta(\rho)=c_1\Gamma(\rho)^{2/3}A(\rho)^{-1}\rho.$$

Universal Airy-one scaling conjecture. There exist universal constants $c_1$ and $c_2$ such that

$$\lim_{t\to\infty}\frac{X_{u\theta(\rho)t^{2/3}}(t)+u\theta(\rho)t^{2/3}/\rho-J(\rho)t/\rho}{-(c_2\Gamma(\rho)t)^{1/3}/\rho}={\cal A}_1(u).$$

This predicts the Airy-one process as the universal limit of the rescaled particle-position process in the flat-density setting. The source presents it as a universality prediction under the stated invariant-measure assumption, with no general proof claimed.