Diameter for permutations of bounded support


Let $ S\subset \textrm{Sym}(n)$ be a set of permutations each of which is of bounded support, that is, each $ \sigma\in S$ moves $ O(1)$ elements of $ \{1,2,\dotsc,n\}$ . Let $ \Gamma$ be the graph whose set of vertices is the set of ordered pairs $ (i,j)$ , $ 1\leq i,j\leq n$ , $ i\ne j$ , and whose edges are $ ((i,j), (i,j)^\sigma)$ , $ \sigma\in S$ . (In other words, $ \Gamma$ is a Schreier graph.) Assume $ \Gamma$ is connected, i.e., $ S$ generates a $ 2$ -transitive group.

Must it be the case that $ \textrm{diam}(\Gamma) = O(n)$ ? Could it be the case that $ \textrm{diam}(\Gamma)\gg n^{1+\delta}$ , $ \delta>0$ , or even $ \delta=1$ ?

What are the answers to these questions if we assume that $ \langle S\rangle$ is all of $ \textrm{Alt}(n)$ or $ \textrm{Sym(n)}$ ?