The deterministic limiting velocity conjecture for stationary ergodic random environments

Let $(X_n)_{n\geq 0}$ be a random walk in a stationary ergodic environment on $\mathbb{Z}^d$. A deterministic limiting velocity is a vector $v$ such that

$$\frac{X_n}{n}\to v$$

for almost every realisation of the environment. The deterministic limiting velocity conjecture. There exists a deterministic vector $v$ such that

$$\frac{X_n}{n}\to v$$

for almost every realisation of the environment. The paper presents this as the law-of-large-numbers problem for stationary ergodic random walks in random environments and claims to establish it affirmatively under additional assumptions; the unrestricted formulation is the motivating conjecture.