The asymptotic covering constant for permutations

Let $S_n$ be the set of permutations of $\{1,\ldots,n\}$, and let $\kappa_n$ denote the minimum number of $(n+1)$-permutations needed to cover every $n$-permutation. Asymptotic covering-constant conjecture. For some constant $A$,

$$\limsup_{n\to\infty}\frac{\kappa_n}{(n+1)!/n^2}=A,$$

and possibly $A\le 2$. The preceding argument gives an upper bound of the same order, while the hypergraph formulation suggests that the covering number has a well-defined asymptotic constant; determining this constant, and in particular whether it is at most $2$, remains open.