This post is a sequel of Diameter of symmetric group.
Let $\Sigma$ a generating subset of $S_n$, $\Gamma(S_n, \Sigma)$ the Cayley graph and $d_{\Sigma}$ the diameter of $\Gamma(S_n, \Sigma)$.
Let $s_n = min_{\Sigma}(\vert \Sigma \vert \times d_{\Sigma})$.
Question: What's the asymptotic of $s_n$ (or the best conjecture for that)?