Nikolov–Segal map conjecture for right exact localizations

Let $L$ be a right exact localization on the category of groups, and let $G$ be a nilpotent group. A homomorphism $f:H\to G$ is a Nikolov–Segal map if

$$[G,G]=[G,\operatorname{Im}(f)].$$

Here $[G,G]$ is the commutator subgroup and $[G,\operatorname{Im}(f)]$ is generated by commutators with one entry in the image of $f$. Nikolov–Segal map conjecture. For any right exact localization $L$ on the category of groups and any nilpotent group $G$, the map $G\to LG$ is a Nikolov–Segal map. The source presents this as a new conjecture after noting that the analogous abelian-cokernel claim fails in general, while it holds for the cited class of right exact localizations in the nilpotent setting; its general status remains open.