We consider the randomly biased random walk on trees in the slow movement regime as in [HS16], whose potential is given by a branching random walk in the boundary case. We study the heavy range up to the $n$-th return to the root, i.e., the number of edges visited more than $k_n$ times. For $k_n=n^\theta$ with $\theta\in(0,1)$, we obtain the convergence in probability of the rescaled heavy range, which improves one result of [AD20].