Locally finite periodic groups saturated with finite simple orthogonal groups of odd dimension

Suppose that $n$ is an odd integer, $n\geq 5$. We prove that a periodic group $G$, saturated with finite simple orthogonal groups $O_n(q)$ of odd dimension over fields of odd characteristic, is isomorphic to $O_n(F)$ for some locally finite field $F$ of odd characteristic. In particular, $G$ is locally finite and countable.