Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions d ≥ 2. In 2002, Sznitman introduced for each $${\gamma\in (0, 1)}$$ the ballisticity conditions (T) γ and (T′), the latter being defined as the fulfillment of (T) γ for all $${\gamma\in (0, 1)}$$ . He proved that (T′) implies ballisticity and that for each $${\gamma\in (0.5, 1)}$$ , (T) γ 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 γ d is a dimension dependent constant taking values in the interval (0.366, 0.388), (T) γ is equivalent to (T′). This is achieved by a detour along the effective criterion, the fulfillment 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.