Sznitman's conjecture that condition $(T')$ is equivalent to ballisticity

Let $P$ be the law of a random walk in an i.i.d. random environment on $\mathbb{Z}^{d}$, and let $\ell_{0}\in S^{d-1}$ be a direction. For $0<\gamma\leq 1$, condition $(T_{\gamma})$ in direction $\ell_{0}$ means that for every $\ell\in S^{d-1}$ in some neighborhood of $\ell_{0}$, there is a finite constant $C$ such that

$$\mathbb{P}^{0}\left(T_{L}^{(-\ell)}<T_{L}^{(\ell)}\right)<Ce^{-L^{\gamma}}.$$

Condition $(T')$ means that $(T_{\gamma})$ holds for some $\frac{1}{2}<\gamma<1$. The walk is ballistic when its limiting velocity $\mathbb{v}$ is nonzero.

Sznitman's conjecture. Condition $(T')$ is equivalent to ballisticity.

Sznitman proved that condition $(T')$ implies ballisticity and that, under this assumption, $(T')$ holds in every direction having positive inner product with the limiting velocity. The converse implication, that ballisticity implies condition $(T')$, is the conjectural part.