The EKR conjecture for unions of length-2 paths

Let $r \leq n$, and let $G$ be the vertex-disjoint union of $n$ paths each of length $2$. A family of independent $r$-sets of $G$ is intersecting if any two of its members have a common vertex, and $G$ is $r$-EKR when every such family has size at most the largest star. The EKR conjecture for unions of length-2 paths. If $r \leq n$, then $G$ is $r$-EKR. The paper has proved this property under a stronger restriction on $r$ and conjectures that the bound $r \leq n$ is sufficient.