Extremal $t$-intersecting Families of Permutations for Large $t$

A set of permutations of $\{1,2,\dots ,n\}$ is $t$-intersecting if any two permutations agree on at least $t$ inputs. A recent work by Kupavskii, in the spirit of the Erdős-Ko-Rado Theorem, shows that for all $t\le n-O\left(\frac{n\log \log n}{\log n}\right)$, every $t$-intersecting family of permutations of $\{1,2,\dots ,n\}$ with the maximum size must be isomorphic to the set $$A_k=\{\sigma :\sigma (i)=i\text{ for at least }t+k\text{ indices }i\in \{1,2,\dots ,t+2k\}\}$$ for some $k$. By refining Kupavskii’s spread approximation technique, we prove that this conclusion holds for a wider range of $t\le n-n^{5\mathrm{/}7+\epsilon }$.