We consider a finite, connected and simple graph $Γ$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(Γ)$, the local action $G_x^{Γ(x)}$ is the action of $\mathrm{Sym}(n)$ on ordered pairs, we show that the group $G_x^{[3]}$, the pointwise stabiliser of a ball of radius three around $x$, is trivial.