Miller's conjecture on odd-prime Miller groups

Let $p$ be an odd prime, and let $G$ be a Miller $p$-group, meaning a finite $p$-group whose automorphism group is abelian. A finite $p$-group $G$ is special when $Z(G)=G'=\Phi(G)$.

Miller's conjecture. If $G$ is a Miller $p$-group for an odd prime $p$, then $G$ is special.

This conjecture proposes a strong structural restriction on finite $p$-groups with abelian automorphism group. The supplied text gives no evidence that it has been proved or disproved.