Berkovich's conjecture on non-inner automorphisms of finite p-groups

Let $p$ be a prime number and let $G$ be a finite nonabelian $p$-group. An automorphism of $G$ is non-inner if it is not induced by conjugation by an element of $G$.

Berkovich's conjecture. Every finite nonabelian $p$-group admits a non-inner automorphism of order $p$.

This conjecture concerns the existence of automorphisms of the smallest possible nontrivial prime order outside the inner automorphism group. The paper proves it for finite $p$-groups of coclass $4$ and $5$ when $p\geq 5$, and for an additional class of odd-order nonabelian $p$-groups with cyclic center; the general conjecture remains open.