Khukhro's derived-length conjecture for finite p-groups

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

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

This is a structural conjecture for finite $p$-groups with automorphisms of prime-power order. The source records a positive answer when $m=1$, including a subgroup of class at most $2$ with $(p,n)$-bounded index, but does not specify the status in general.