ASEPsc particle-position Tracy–Widom GOE conjecture

Let $X_N^{{\rm ASEPsc}}(t)$ be the position of particle $N$ in ASEPsc with flat initial condition, and use the functions $J(\beta,E)$ and $\Gamma(\beta,E)$ defined by the model parameters. Define

$$X^{\rm resc}_N(t)=\frac{X_N^{{\rm ASEPsc}}(t)-X_N^{{\rm ASEPsc}}(0)-2J(\beta,E)t}{-2\Gamma(\beta,E)^{1/3}t^{1/3}}.$$

ASEPsc particle-position conjecture. For any $N\in\mathbb{Z}$ and every $s\in\mathbb{R}$,

$$\lim_{t\to\infty}\mathbb{P}\left(X^{\rm resc}_N(t)\le s\right)=F_{\rm GOE}(2s),$$

where $F_{\rm GOE}$ is the Tracy–Widom GOE distribution. This predicts the GOE fluctuation law for particle positions in ASEPsc under KPZ $t^{1/3}$ scaling; the source does not state that it has been proved.