  We are interested in the randomly biased random walk on the supercritical
Galton--Watson tree. Our attention is focused on a slow regime when the biased
random walk $(X_n)$ is null recurrent, making a maximal displacement of order
of magnitude $(\log n)^3$ in the first $n$ steps. We study the localization
problem of $X_n$ and prove that the quenched law of $X_n$ can be approximated
by a certain invariant probability depending on $n$ and the random environment.
As a consequence, we establish that upon the survival of the system,
$\frac{|X_n|}{(\log n)^2}$ converges in law to some non-degenerate limit on
$(0, \infty)$ whose law is explicitly computed.
