Farjoun's abelian-cokernel conjecture for group localizations

Let $L$ be a localization on the category of groups, let $G$ be a group, and let $G\to LG$ be the localization map. The cokernel of this map is the quotient of $LG$ by the normal closure of the image of $G$. Farjoun's abelian-cokernel conjecture. For any localization $L$ on the category of groups and any group $G$, the cokernel of the map $G\to LG$ is abelian. The source proves that this conjecture fails for the Baumslag rationalization and the free group of rank $2$, so the conjecture is refuted.