On the conjugacy separability in the class of finite $p$-groups of   finitely generated nilpotent groups
  It is proved that for any prime $p$ a finitely generated nilpotent group is conjugacy separable in the class of finite $p$-groups if and only if the torsion subgroup of it is a finite $p$-group and the quotient group by the torsion subgroup is abelian. 
E. A. Ivanova