ASEP backward-geodesic endpoint convergence conjecture

Let $X_n(t)$ denote the particle positions in ASEP with flat initial condition, and let $N(t\downarrow 0)$ be the index associated with the backward geodesic endpoint. For jump rates $p,q$, define

$$B^{\rm ASEP}_t=\frac{X_{N(t\downarrow0)}(0)-X_N(0)-(p-q)t/2}{2^{1/3}((p-q)t)^{2/3}}.$$

ASEP backward-geodesic endpoint conjecture. For any $N\in\mathbb{Z}$,

$$\lim_{t\to\infty}B^{\rm ASEP}_t\stackrel{d}{=}\hat u.$$

Here $\hat u$ is the limiting endpoint distribution for the corresponding point-to-point last-passage-percolation geodesic. This is a KPZ-scaling prediction for ASEP, following the proved analogous result for TASEP; its resolution is not specified in the source.