Normalized root words and reflective power-subgroup completion

Fix a prime p, and let Pₙ(G) be the subgroup generated by the pⁿ-th powers in a group G. We consider the full category of groups admitting a two-variable word w and an integer r≥1 such that w(a,b)ᵖ=aᵖ bpʳ and $w(a,1)=a$. Every nilpotent group belongs to this category: for each positive class bound c, one may take $r=c$ and one word valid in all groups of class at most c. For G=ₙ G/Pₙ(G) we prove the exact equality Pₙ( G)=( G→ G/Pₙ(G)). Consequently the inverse-limit and intrinsic power-subgroup topologies coincide, and completion is a reflector onto the complete Hausdorff objects. This gives a categorical setting of the kind requested in Kourovka Problem 10.52.