The ASEP almost-sure speed convergence conjecture

Let $pin(1/2,1]$ and write $\rho=2p-1$. Consider the asymmetric simple exclusion process (ASEP), with particles jumping right at rate $p$ and left at rate $1-p$, and let $X_0(t)$ denote the position at time $t$ of the particle initially labelled $0$. The ASEP speed convergence conjecture. In the ASEP,

$$\lim_{t\to\infty} \frac{X_0(t)}{t}$$

exists almost surely, and its distribution is uniform on $[-\rho,\rho]$. This is equivalent to the existence of the ASEP speed-process measure $\mu^{\mathrm{ASEP}}$, which is translation invariant and has each speed $U_n$ uniformly distributed on $[-\rho,\rho]$. Distributional convergence is known by the methods of Ferrari and Kipnis, but almost-sure convergence remains unproved in the source.