Let $G$ be a (possibly infinite) periodic group, and suppose that $G$ admits a maximal finite $p$-subgroup $P$. By this I mean that we do not assume that $P$ is not strictly contained in any other $p$-subgroup of $G$, but that it is not contained in any other finite $p$-subgroup of $G$.
Does it follow that all maximal finite $p$-subgroups of $G$ are conjugate? If not, what are some reasonable assumptions one might add for this to hold?