Fuentes–Kamat conjecture on Erdős–Ko–Rado subfamilies of perfect matchings

Let $M_n$ be the graph with vertex set

$$V(M_n)=\{a_1, a_2, \dots, a_n, b_1, b_2,\dots, b_n\}$$

and edge set

$$E(M_n)=\{(a_1,b_1), (a_2,b_2), \dots, (a_n,b_n)\}.$$

Let $\mathcal{H}^{(p,s)}(n)$ be the family of subsets of $V(M_n)$ that contain exactly $2p+s$ vertices and span exactly $p$ edges, equivalently, that contain $p$ disjoint edges and $s$ isolated vertices. For a family $\mathcal{H}$ and an element $x$, write $\mathcal{H}_x=\{H\in\mathcal{H}:x\in H\}$; the family $\mathcal{H}$ is EKR if some $x$ satisfies $|\mathcal{F}|\leq|\mathcal{H}_x|$ for every intersecting subfamily $\mathcal{F}\subseteq\mathcal{H}$. Fuentes–Kamat conjecture. For non-negative integers $p,s$ satisfying $1\leq 2p+s\leq n$, $\mathcal{H}^{(p,s)}(n)$ is EKR. This conjecture concerns an Erdős–Ko–Rado property for subfamilies of subsets associated with a perfect matching; the source notes that Fuentes and Kamat had previously conjectured this statement and proved it for a certain range of parameters, while the general assertion is presented here as the conjectural claim.