Universal backward-geodesic endpoint argmax conjecture

Consider an exclusion process with generator specified by the paper and initial condition $X_n(0)=-\lfloor n/\rho\rfloor$ for some $\rho\in(0,1)$. Assume its translation-invariant stationary measure $\mu_\rho$ satisfies the paper's Assumption~. Let $J(\rho)$ be the stationary current and let $A(\rho)$ and $\Gamma(\rho)$ be the associated KPZ parameters. Let $N(t\downarrow0)$ denote the backward-geodesic endpoint index, and define

$$\hat u=\operatorname*{argmax}_{u\in\mathbb{R}}\left\{{\cal A}_2(u)-u^2\right\}.$$

Universal backward-geodesic endpoint conjecture. There exists a universal constant $c_2$ such that

$$\lim_{t\to\infty}\frac{X_{N(t\downarrow0)}(0)-X_N(0)-(J(\rho)/\rho-J'(\rho))t}{2^{1/3}\Gamma(\rho)^{2/3}A(\rho)^{-1}t^{2/3}}\stackrel{d}{=}\hat u.$$

This predicts the universal argmax distribution of the Airy-two process minus a parabola for backward-geodesic endpoints. The source motivates the scale using KPZ theory and the known TASEP result, but gives no general proof.