Ahlswede–Khachatrian-type conjecture for t-intersecting families of permutations

Let $n$ and $t$ be positive integers, let $cSigma_n$ be the collection of permutations of $[n]$, and let $cA_k$ be the family

$$cA_k=\{\sigma\incSigma_n:\sigma(i)=i\text{ for at least }t+k\text{ indices }i\in[t+2k]\}.$$

A family $\calF\subseteqcSigma_n$ is $t$-intersecting if any two of its permutations agree on at least $t$ values of $i$. Ahlswede–Khachatrian-type conjecture. If $\calF\subseteqcSigma_n$ is a $t$-intersecting family, then

$$|\calF|\leq\max_k|\calA_k|.$$

Moreover, if equality holds, then $\calF=\sigma\calA_k\tau$ for some $k$ and permutations $\sigma$ and $\tau$. This is the permutation analogue of the Ahlswede–Khachatrian theorem for $t$-intersecting families of sets. The conjecture asks both for the sharp maximum size and for a classification of all extremal families; its resolution is not specified in the supplied source.