Conjecture on the displacement of a biased random walk on a tree

Let $(X_n)_{n\geq 0}$ be the random walk in a random environment on a tree, let $|X_n|$ denote the distance of the walk from the root, and let $\mathbb{P}$ be the governing probability measure. Assume

$$and$$

. Displacement conjecture. Under $\mathbb{P}$, on the set of non-extinction, $\frac{|X_n|}{(\log n)^2}$ converges weakly to a limit law which is finite and strictly positive. This conjecture concerns the position of the walk at time $n$, whose order of magnitude is stated to be unknown, while the maximum displacement up to time $n$ has order of magnitude $(\log n)^3$. The source presents the claim as an expectation and gives no resolution.