  In this paper we study the recurrence and transience of the
$\mathbb{Z}^d$-valued branching random walk in random environment indexed by a
critical Bienaym\'e-Galton-Watson tree, conditioned to survive. The environment
is made either of random conductances or of random traps on each vertex. We
show that when the offspring distribution is non degenerate with a finite third
moment and the environment satisfies some suitable technical assumptions, then
the process is recurrent up to dimension four, and transient otherwise. The
proof is based on a truncated second moment method, which only requires to have
good estimates on the quenched Green's function. Notably, this provides an
alternate proof to the previous argument of Benjamini and Curien in the case of
a deterministic, homogeneous environment.
