The law of large numbers for the ballistic random walk

Let $(Z_n)_{n\geq 0}$ be the random walk defined in Section 2, and let $\mathbb{P}$ denote its probability measure. Law of large numbers conjecture. There exists $\Delta\in\mathbb{R}^2$ such that

$$\mathbb{P}\text{-almost surely},\qquad \frac{Z_n}{n}\xrightarrow[n\to\infty]{}\Delta.$$

This would establish a deterministic asymptotic direction and velocity for the ballistic random walk in the considered two-dimensional random environment. The conjecture is presented as the next step toward a complete law of large numbers, and its resolution is not supplied here.