Conjecture on the potential energy 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 $V(X_n)$ denote its potential energy, and let $\mathbb{P}$ be the governing probability measure. Assume

$$and$$

. Potential-energy conjecture. Under $\mathbb{P}$, on the set of non-extinction, $\frac{V(X_n)}{\log n}$ converges weakly to a limit law which is finite and strictly positive. This conjecture predicts that the potential energy at time $n$ has logarithmic scale, much smaller than the maximum potential energy up to time $n$, whose almost-sure growth is $\frac12(\log n)^2$ under the stated assumptions. The source presents this as an expectation, and gives no resolution.