Let $\Sigma_n\subset G$ be a set of generators of the symmetric group $S_n$. It is a well-known conjecture that the diameter of the Cayley graph $\Gamma(S_n,\Sigma_n)$ is at most $n^C$ for some absolute constant $C$. (The diameter of the Cayley graph is just the maximum of $\ell(g)$ for $g\in S_n$, where $\ell(g)$ is the length of the shortest word on $A \cup A^{-1}$ equal to $g$.)

For $\Sigma_n$ of bounded size, the diameter cannot be less than a constant times $\log |S_n|$, i.e., a constant times $n\log n$.

It is clear and well-known that, for $\Sigma_n = \{(1 2),(1 2 \dotsb n)\}$, the diameter of $\Gamma(S_n, \Sigma_n)$ is at least a constant times $n^2$. (It is also at most that.)

Are there any examples of generating sets $\Sigma_n$ for which the diameter is larger than $n^{2+\epsilon}$ for every (or infinitely many) $n$? Larger than $n^2 (\log n)^A$ for some $A>0$ and infinitely many $n$?