The full Erdős–Ko–Rado conjecture for t-intersecting perfect matchings

Let $K_{2n}$ be the complete graph on $2n$ vertices, and let $\mathcal{F}$ be a family of its perfect matchings. The family is $t$-intersecting if $|m\cap m'|>t$ for all $m,m'\in\mathcal{F}$. It is trivially $t$-intersecting when, for some collection $T$ of $t$ pairwise disjoint 2-sets of $[2n]$,

$$\mathcal{F}_T=\{m\in\mathcal{M}_{2n}:T\subseteq m\}.$$

The full Erdős–Ko–Rado conjecture for t-intersecting perfect matchings. If $\mathcal{F}$ is a $t$-intersecting family of perfect matchings of $K_{2n}$, then

$$|\mathcal{F}|\leq (2(n-t)-1)!!.$$

Moreover, equality holds if and only if $\mathcal{F}$ is trivially $t$-intersecting. This extends the known Erdős–Ko–Rado theorem for intersecting families of perfect matchings and remains an open generalization for $t$-intersecting families.