Aldous's scaling-limit conjecture for the stationary East process

Let the stationary East process be defined on the half-line $[0,+\infty)$, with vacancy density $q$, and let $X_t$ denote the rescaled process obtained by rescaling space by $q$ and speeding up time by the relaxation time. A Poisson process of rate $1$ is a point process whose particles are distributed according to a homogeneous Poisson point process with intensity $1$.

Aldous's conjecture. As $q\downarrow 0$, the rescaled stationary East process converges to a limit point process $X_t$ on $[0,+\infty)$ such that:

1. At each fixed time $t$, $X_t$ is a Poisson process of rate $1$ of particles, equivalently, $0$ spins.
2. For every $\ell>0$, each particle, at a positive rate depending on $\ell$, deletes all particles to its right up to distance $\ell$ and replaces them by a new Poisson process of rate $1$.

The conjecture describes the expected scaling limit at the equilibrium length scale. The paper proves that time-scale separation fails at scales $L\;\geqslant\; 1/q$, so the scaling limit of the stationary East process on those scales remains open; the proposed limit is presented as consistent with the rigorous results.