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

Let $K_{2n}$ be the complete graph on $2n$ vertices, and let $\mathcal{F}$ be a family of perfect matchings of $K_{2n}$. The family is $t$-intersecting if any two of its members share at least $t$ edges, and it is canonically $t$-intersecting if every member contains a fixed set of $t$ disjoint edges of $K_{2n}$. The Erdős–Ko–Rado conjecture. For $t\in\mathbb{N}$ and sufficiently large $n$, if $\mathcal{F}$ is $t$-intersecting, then

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

Moreover, equality holds if and only if $\mathcal{F}$ is canonically $t$-intersecting. This is the conjectured nonbipartite analogue of the Erdős–Ko–Rado theorem for perfect matchings of $K_{n,n}$; the statement concerns both the extremal size and the characterization of all equality cases.