  We study the strong form of the ballistic conjecture for random walks in
random environments (RWRE). This conjecture asserts that any RWRE which is
directionally transient for a nonempty open set of directions satisfies
condition $(T)$ (annealed exponential decay for the unlikely exit probability).
Specifically, we introduce a ballisticity condition which is fulfilled as soon
as a polynomial condition of degree greater than $d-1$ holds. Under that
hypothesis we prove condition $(T)$, which turns this condition into the
weakest-known ballisticity assumption. We recall that standard arguments to
prove that a ballisticity condition implies directional transience require at
least polynomial decay greater than degree $d$. Furthermore, in the one
dimensional case we provide an alternative proof which proves the equivalence
between transient behaviour and annealed arbitrary decay for the unlikely exit
probability, we expect that this new argument might be used in higher
dimensions.
