Large Ree groups acting point and line-primitively on finite generalised quadrangles

Suppose we have a finite thick generalised quadrangle whose automorphism group $G$ acts primitively on both the set of points and the set of lines. Then $G$ must be almost simple. Furthermore, $\textrm{soc}(G)$ cannot be isomorphic to ${}^2 B_2(2^{2m+1})$ or ${}^2 G_2(3^{2m+1})$, where $m$ is a positive integer. In this paper, we show that $\textrm{soc}(G)$ cannot be isomorphic to ${}^2 F_4(2^{2m+1})$.