Sharp stability for cross $t$-intersecting families of permutations in the linear range

Two families $\mathcal{F},\mathcal{G}\subseteq S_n$ are cross $t$-intersecting if every $σ\in\mathcal{F}$ and $τ\in\mathcal{G}$ agree on at least $t$ points. A $t$-coset is a coset of the stabilizer of $t$ points. A subset of $S_n$ is non-trivial if it is not contained in any $t$-coset. Let $d_m$ de