The solution of the restricted Burnside problem is based on the Hall - Higman reduction, the classification of finite simple groups and a celebrated result of Efim Zelmanov on finding an upper bound for nilpotency class of Lie algebras satisfying the linearized Engel condition of length $n$ in terms of the number of generators of the Lie algebra and $n$.
Zelmanov's result has a consequence for finite $p$-groups: There exists a function $f:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}$ such that $|G|\leq f(p^e,d)$ for any finite $d$-generator $p$-group $G$ of exponent dividing $p^e$.
Is the above group theoretic consequence of the Zelmanov's deep result equivalent to the Zelmanov's Lie algebra one?
