Ellis–Friedgut–Pilpel conjecture on extremal t-intersecting families of permutations

Let $S_n$ be the symmetric group, and let $\mathrm{Fixed}(\sigma)$ denote the set of fixed points of $\sigma$. A family $F\subset S_n$ is $t$-intersecting if any two of its permutations agree on at least $t$ points. For $n,t,r$ with $n\in\mathbb N$, $1\leq t\leq n$, define

$$\mathcal F_{n,t,r}=\{\sigma\in S_n:|\mathrm{Fixed}(\sigma)\cap\{1,2,\ldots,t+2r\}|\geq t+r\}.$$

Ellis–Friedgut–Pilpel conjecture. For any $n\in\mathbb N$ and $1\leq t\leq n$, the maximum size of a $t$-intersecting family in $S_n$ is attained by one of the families $\mathcal F_{n,t,r}$. In particular, for all $t<n/2$, the maximum is attained by the $t$-umvirate

$$\mathcal F_{n,t,0}=\{\sigma\in S_n:\mathrm{Fixed}(\sigma)\supset [t]\}.$$

Furthermore, every maximum-sized $t$-intersecting family is a double translate $\tau_1\mathcal F_{n,t,r}\tau_2$ for some $r$ and some $\tau_1,\tau_2\in S_n$. The source presents this as a daring conjecture after noting earlier partial results; the supplied status is unknown.