The Burness-Giudici Conjecture on Primitive Groups with Socle $Ree(q)$ and $Sz(q)$

Let $G$ be a transitive permutation group on $\mathrm{\Omega }$ containing two points $\alpha ,\beta$ such that $G_{\alpha }\cap G_{\beta }=1$. The Saxl graph $\mathrm{\Sigma }(G)$ of $(G,\mathrm{\Omega })$ is defined as the graph with vertex set $\mathrm{\Omega }$, where two vertices ${\alpha }^{\mathrm{\prime }},{\beta }^{\mathrm{\prime }}$ are adjacent if and only if $G_{{\alpha }^{\mathrm{\prime }}}\cap G_{{\beta }^{\mathrm{\prime }}}=1$. Burness and Giudici conjectured that for any primitive permutation group $G$, its Saxl graph $\mathrm{\Sigma }(G)$ satisfies the property that any two vertices share a common neighbor. We focused on proving this conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$; that is, groups with $soc(G)\in \{PSL(2,q),PSU(3,q),Ree(q),Sz(q)\}$. The case $soc(G)=PSL(2,q)$ has been published in two papers. In this paper, we treat the cases where $soc(G)\in \{Ree(q),Sz(q)\}$.