Holroyd–Talbot conjecture for pendant path graphs

Let $P_n^*$ denote the pendant path graph obtained by appending a pendant edge to each vertex of the path $P_n$. An $r$-star is the family of independent $r$-sets containing a fixed vertex, and a graph is $r$-EKR when an $r$-star has maximum size among all intersecting families of independent $r$-sets.

Pendant-path conjecture. The pendant path graph $P_n^*$ is $r$-EKR whenever

$$n \geq 2r.$$

The conjecture is proposed as a case of the Holroyd–Talbot conjecture. The paper proves that $P_n^*$ is not $n$-EKR for $n\geq 4$, which lies outside the conjectured range $n\geq 2r$; the stated pendant-path conjecture remains open in the supplied text.