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My answer to your question here included a proof of this. Indeed, such a group has a non-abelian 2-nilpotent quotient. In the discrete case, it even shows that there is a non-abelian torsion-free 2-nilpotent quotient. In the profinite case, it shows that for all $p$ (although I gave a full proof only for odd $p$) there is a non-abelian 2-nilpotent $p$-group ...

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