Equivalence of transience and ballisticity in random environment

Let $d\ge 2$ and let $l\in\mathbb S^{d-1}$. A random walk in a uniformly elliptic i.i.d. random environment is transient in the direction $l$ if

$$\lim_{n\to\infty}X_n\cdot l=\infty$$

$P_0$-almost surely, and it is ballistic in the direction $l$ if

$$\liminf_{n\to\infty}\frac{X_n\cdot l}{n}>0$$

$P_0$-almost surely. Transience-ballisticity conjecture. Transience in the direction $l$ implies ballisticity in the direction $l$. Whether this implication holds in all dimensions $d\ge 2$ remains open; in dimension one, transience need not imply ballisticity.