Low rank groups of Lie type 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. In this paper, we show that $\mathrm{soc}(G)$ cannot be isomorphic to $\mathrm{Sz}(2^{2m+1})$ or $\mathrm{Ree}(3^{2m+1})$ where $m$ is a positive integer.