  We prove that the dimension drop phenomenon holds for the harmonic measure
associated to a transient random walk in a random environment (as defined by R.
Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on an
infinite Galton-Watson tree without leaves. We use regeneration times and
ergodic theory techniques from the work of R. Lyons, R. Pemantle and Y. Peres
in 1996 to give an explicit construction of the invariant measure for the
forward environment seen by the particule at exit times which is absolutely
continuous with respect to the joint law of the tree and the path of the random
walk.
