Backward-geodesic discrepancy limiting-distribution conjecture

Let $D_N(t)$ be the discrepancy between the backward-geodesic endpoint index $N(t\downarrow0)$ and the corresponding minimization index in the exclusion-process setting considered in the paper. Backward-geodesic discrepancy conjecture. There exists a non-degenerate discrete distribution $G$ such that, for every $s\in\mathbb{R}$,

$$\lim_{t\to\infty}\mathbb{P}\left(D_N(t)\le s\right)=G(s).$$

The preceding discussion notes that the scaled discrepancy is expected to vanish, while the unscaled discrepancy may converge to a non-degenerate random variable. The claim is supported by numerical simulation, but no proof or resolution is given.