Farjoun's finite p-group surjectivity conjecture for group localizations

Let $L$ be a localization on the category of groups, and let $G$ be a finite $p$-group, meaning a finite group whose order is a power of the prime $p$. Write $G\to LG$ for the localization map. Farjoun's finite $p$-group surjectivity conjecture. For any localization $L$ on the category of groups and for any finite $p$-group $G$, the map $G\to LG$ is surjective. This conjecture was formulated by Emmanuel Dror Farjoun in personal communication; its status is not resolved in the supplied source.