Kézdy–Snevily conjecture for the covering function of permutation space

Let $\mathcal{S}_n$ be the set of permutations of $[n]=\{1,2,\dots,n\}$, with Hamming distance $d_H$. Define $f(n,s)$ to be the minimum size of a subset of $\mathcal{S}_n$ having covering radius at most $n-s$. Kézdy–Snevily conjecture. If $n$ is even, then $f(n,2)=n$; if $n$ is odd, then $f(n,2)>n$. The conjecture is motivated by its implications for the latin-square transversal conjectures, and the paper proves that its odd-$n$ case implies the whole conjecture. The source gives no resolution status for the conjecture itself.