Aldous–Diaconis continuum-limit conjecture for East-model excursions

Let ${\bf X}^0(t)$ be the East process on the positive integers, started from a single particle at site $0$. Let $r({\bf x})$ be the rightmost occupied site in configuration ${\bf x}$. Let $p$ be the particle-density parameter, and let $\omega(p)$ decrease to zero sufficiently slowly. Aldous–Diaconis continuum-limit conjecture. There exist constants $t(p)$ and a function $G(x)$ for $0<x<\infty$ such that, for every such $\omega(p)$,

$$P\left(\sup_{0\leq s\leq \omega(p)t(p)}r({\bf X}^0(s))>\frac{x}{p}\right)\sim G(x)\omega(p)$$

as $p\downarrow0$, with $x$ fixed. This conjecture describes the short-time probability of an East-process excursion reaching macroscopic distance on the scale $1/p$. The supplied text states the conjecture but gives no evidence of resolution, so its status is left open.