The transience implies ballisticity conjecture for random walk in random environment

Let $d\geq 2$. Assume that the environment is independent and identically distributed and uniformly elliptic: there is an ellipticity constant $\kappa>0$ such that the transition probabilities are uniformly bounded below by $\kappa$. A random walk in random environment that is transient with respect to every direction in an open subset of $\mathbb S^{d-1}$ is called transient in that subset.

Transience–ballisticity conjecture. Under these assumptions, RWRE that is transient with respect to all directions in an open subset of $\mathbb S^{d-1}$ is necessarily ballistic.

This is a fundamental open problem in dimensions at least two. The paper describes partial progress through conditions $(T)_\gamma$ and the effective criterion, but does not establish the conjecture in full.