  Consider a random walk in a uniformly elliptic i.i.d. random environment in
dimensions $d\ge 2$. In 2002, Sznitman introduced for each $\gamma\in (0,1)$
the ballisticity conditions $(T)_\gamma$ and $(T'),$ the latter being defined
as the fulfilment of $(T)_\gamma$ for all $\gamma\in (0,1).$ He proved that
$(T')$ implies ballisticity and that for each $\gamma\in (0.5,1),$ $(T)_\gamma$
is equivalent to $(T')$. It is conjectured that this equivalence holds for all
$\gamma\in (0,1).$ Here we prove that for $\gamma\in (\gamma_d,1),$ where
$\gamma_d$ is a dimension dependent constant taking values in the interval
$(0.366,0.388),$ $(T)_\gamma$ is equivalent to $(T').$ This is achieved by a
detour along the effective criterion, the fulfilment of which we establish by a
combination of techniques developed by Sznitman giving a control on the
occurrence of atypical quenched exit distributions through boxes.
