Khukhro's class-bounded subgroup conjecture for finite p-groups

Let $G$ be a finite $p$-group admitting an automorphism of order $p$ with $p^m$ fixed points. A subgroup has $m$-bounded class if its nilpotency class is bounded by a function of $m$, and an index is $(p,m)$-bounded if it is bounded by a function of $p$ and $m$.

Khukhro's conjecture. $G$ contains a subgroup of $m$-bounded nilpotency class and $(p,m)$-bounded index.

The conjecture concerns structural consequences of a fixed-point-free-type automorphism condition on finite $p$-groups. The source records a positive answer when $m=1$, while its general status is not specified here.