Sznitman's transience implies ballisticity conjecture for uniformly elliptic RWRE

Let $d\ge 2$, let $l\in\mathbb S^{d-1}$, and consider a random walk in a uniformly elliptic i.i.d. environment on $\mathbb Z^d$. The walk is transient in direction $l$ when

$$P_0(A_l)=1,$$

where

$$A_l:=\left\{\lim_{n\to\infty}X_n\cdot l=\infty\right\}.$$

It is ballistic in direction $l$ when

$$P_0\text{-almost surely},\qquad \liminf_{n\to\infty}\frac{X_n\cdot l}{n}>0.$$

Sznitman's transience-ballisticity conjecture. If the walk is transient in direction $l$, then it is ballistic in direction $l$.

This conjecture asks whether directional transience alone implies positive asymptotic speed for uniformly elliptic i.i.d. random walks in random environment. It was stated in the context of extending ballisticity criteria beyond presently known sufficient conditions; its resolution status is not specified in the supplied text.