Large-degree distinguishing-number conjecture for symmetric-group actions

Let $S_n$ act on a set, and let its distinguishing number be the least number of labels needed to eliminate all nonidentity elements of the action. The large-degree distinguishing-number conjecture. There exists an integer $N$ such that, for every $n>N$, the symmetric group $S_n$ cannot act with distinguishing number $n-2$. This is motivated by the fact that for $n>4$, $A_n$ is the only nontrivial normal subgroup of $S_n$ and by non-tight bounds in the construction of pseudocliques.