Ellis's setwise intersecting family conjecture for the symmetric group

Let $n$ and $t$ be positive integers with $n e 0$ and $n\geq t$. A family $\mathcal{F}\subset\operatorname{sym}(n)$ is $t$-setwise intersecting if every two permutations in $\mathcal{F}$ agree on some $t$-\subset of $\{1,2,\ldots,n\}$ setwise. A setwise stabilizer of a $t$-\subset $S$ is the subgroup of permutations that map $S$ to itself.

Ellis's conjecture. If $\mathcal{F}$ is a $t$-setwise intersecting family of $\operatorname{sym}(n)$, then

$$|\mathcal{F}|\leq t!(n-t)!.$$

Moreover, if $(n,t)\notin\{(4,2),(5,2)\}$ and equality holds, then $\mathcal{F}$ is a coset of a setwise stabilizer of a $t$-subset of $\{1,2,\ldots,n\}$.

This is an Erdős–Ko–Rado-type extremal and stability statement for setwise intersection in symmetric groups. The exceptional parameter pairs $(4,2)$ and $(5,2)$ are explicitly excluded from the equality characterization; the supplied text gives no resolution status beyond presenting the claim as a conjecture.