Set-wise intersection conjecture for perfect matchings

Let $K_{2k}$ be the complete graph on $2k$ vertices. For $t\leq \lfloor k/2\rfloor$, two perfect matchings are set-wise $t$-intersecting if some $t$ edges in each matching cover the same set of vertices. A family is set-wise $t$-intersecting if every pair of its members is set-wise $t$-intersecting.

Set-wise intersection conjecture. For $k\geq 2t$, the largest set-wise $t$-intersecting family of perfect matchings in $K_{2k}$ has size

$$(2t-1)!!(2k-2t-1)!!.$$

Moreover, every maximum-size family is an orbit of the Young subgroup $Sym(2t)\times Sym(2k-2t)$.

This conjecture extends the Erdős–Ko–Rado theorem from ordinary intersection to set-wise intersection. The paper proves the corresponding result for $t=2$; the cases $t\geq 3$ remain open.