Cameron–Ku stability conjecture for intersecting families of permutations

Let $G$ be the graph whose vertices are the elements of the symmetric group $S_n$, with two vertices $\pi$ and $\pi'$ adjacent when $\pi(i)\neq\pi'(i)$ for every $1\leq i\leq n$. An independent set is a set of vertices containing no adjacent pair. For $1\leq i,j\leq n$, let $S_{ij}=\\{\pi\in S_n:\pi(i)=j\\}$; each $S_{ij}$ is an independent set of size $(n-1)!$. Cameron–Ku conjecture. There is a constant $c$ such that every independent set of size at least $c(n-1)!$ is a subset of an independent set of size $(n-1)!$. Cameron and Ku proved that the sets $S_{ij}$ are the only maximum independent sets, and this conjecture asks for a corresponding stability statement for sufficiently large independent sets.