Asymptotic velocity conjecture for rank particles in competing Brownian systems

Fix drift parameters $(g_n)_{n\geq 1}$ and a parameter $a$ as in Theorem~. Consider an infinite system of competing Brownian particles started from

$$X_1(0)=0,$$

and with the initial gaps distributed according to $(Z_k(0))_{k=1}^{\infty}\sim\pi_a$. For a fixed rank $k\in\mathbb{Z}_{>0}$, let $Y_k(t)$ denote the position of the $k$-th rank particle at time $t$. Asymptotic velocity conjecture. Almost surely,

$$\frac{Y_k(t)}{t}\longrightarrow-\frac{a}{2}\qquad\text{as }t\to\infty.$$

This predicts a common deterministic long-time velocity for every fixed rank particle under stationary gap initial data.