Kalikow's 0-1 conjecture for directional escape in random environments

Let $X=(X_n)_{n\geq 0}$ be a random walk in a stationary ergodic environment on $\mathbb{Z}^d$, and let $S^{d-1}$ denote the unit sphere. For a direction $\ell\in S^{d-1}$, define the directional escape event

$$A_{\ell}=\{X_n\mathbin{.}\ell\to+\infty\}.$$

Kalikow's 0-1 conjecture.

$$\mathbb{P}[A_{\ell}]\in\{0,1\}.$$

This asserts that, for each direction, the walk either escapes to infinity in that direction or does not, with probability zero or one. The source states that this conjecture has counterexamples for elliptic ergodic environments, including prior work in two dimensions; it is therefore refuted in the stated generality.