Periodic groups saturated with finite simple groups of Lie type of rank $1$

A group $G$ is saturated with groups from a set $\mathfrak R$ of groups if every finite subgroup of $G$ is contained in a subgroup of $G$ that is isomorphic to some group in $\mathfrak R$. Previously [Kourovka Notebook, Quest. 14.101], the question was posed whether a periodic group saturated with finite simple groups of Lie type whose ranks are bounded in totality is itself a simple group of Lie type.
A partial answer to this question is given for groups of Lie type of rank $1$. We prove the following:
Theorem. Let a periodic group $G$ be saturated with finite simple groups of Lie type of rank $1$. Then $G$ is isomorphic to a simple group of Lie type of rank $1$ over a suitable locally finite field.