Large minimal invariable generating sets in the finite symmetric groups

For a finite group $G$, let $m_I(G)$ denote the largest possible cardinality of a minimal invariable generating set of $G$. We prove an upper and a lower bound for $m_I(S_n)$, which show in particular that $m_I(S_n)$ is asymptotic to $n\mathrm{/}2$ as $n\to \mathrm{\infty }$.