Conjecture that every characteristic subgroup of nilpotent group is powering-invariant

Statement
This is a conjecture. It has not been proved and it may well be false.

The conjecture has the following equivalent formulations:


 * 1) In a nilpotent group, every characteristic subgroup (i.e., characteristic subgroup of nilpotent group) is a powering-invariant subgroup (i.e., powering-invariant subgroup of nilpotent group).
 * 2) Every nilpotent group is a group in which every characteristic subgroup is powering-invariant.

Related conjectures

 * Conjecture that every characteristic subring of nilpotent Lie ring is powering-invariant

Truth based on ambient group

 * Characteristic subgroup of abelian group implies powering-invariant

Truth based on nature of characteristic subgroup

 * Upper central series members are powering-invariant
 * Lower central series members are powering-invariant in nilpotent group
 * Derived series members are powering-invariant in nilpotent group