Deza–Frankl conjecture on maximum k-intersecting families of permutations

Let $S_n$ be the symmetric group. For $k\in\mathbb{N}$, a subset $I\subset S_n$ is $k$-intersecting if any two permutations in $I$ agree on at least $k$ points. A $k$-coset is a set of the form

$$T_{i_1\mapsto j_1,\ldots,i_k\mapsto j_k}=\{\sigma\in S_n:\sigma(i_t)=j_t\ (1\leq t\leq k)\},$$

where the $i_t$ are distinct and the $j_t$ are distinct; it has size $(n-k)!$.

Deza–Frankl conjecture. For any $k\in\mathbb{N}$ and any $n$ sufficiently large depending on $k$, if $I\subset S_n$ is $k$-intersecting, then

$$|I|\leq(n-k)!.$$

Equality holds if and only if $I$ is a $k$-coset of $S_n$.

For small $n$ relative to $k$, the $k$-cosets need not be largest, as the source gives an explicit competing family. The conjecture asserts that the stated bound and equality characterization hold once $n$ is sufficiently large depending on $k$.