Stable fluctuations for randomly biased walks on leafless Galton–Watson trees

Consider randomly biased random walks on Galton–Watson trees without leaves, with exponent $\alpha\in(1,2)$, and assume a non-lattice condition. Stable-fluctuation conjecture. It would be interesting to determine whether stable fluctuations occur in this regime; this should hold only if no slowdown strong enough to break the central limit theorem can occur outside of pipes. The statement is conditional and exploratory, so its precise mathematical formulation remains to be established.