  We consider a random walk in a uniformly elliptic i.i.d. random environment
in $\mathbb Z^d$ for $d\ge 2$. It is believed that whenever the random walk is
transient in a given direction it is necessarily ballistic. In order to
quantify the gap which would be needed to prove this equivalence, several
ballisticity conditions have been introduced. In particular, in 2001 and 2002,
Sznitman defined the so called conditions $(T)$ and $(T')$. The first one is
the requirement that certain unlikely exit probabilities from a set of slabs
decay exponentially fast with their width $L$. The second one is the
requirement that for all $\gamma\in (0,1)$ condition $(T)_\gamma$ is satisfied,
which in turn is defined as the requirement that the decay is like
$e^{-CL^\gamma}$ for some $C>0$. In this article we prove a conjecture of
Sznitman of 2002, stating that $(T)$ and $(T')$ are equivalent. Hence, this
closes the circle proving the equivalence of conditions $(T)$, $(T')$ and
$(T)_\gamma$ for some $\gamma\in (0,1)$ as conjectured by Sznitman, and also of
each of these ballisticity conditions with the polynomial condition $(P)_M$ for
$M\ge 15d+5$ introduced by Berger, Drewitz and Ramirez in 2014.
