Holroyd–Talbot conjecture for independent-set families

Prove that for every graph G and every integer r satisfying 1≤r≤μ(G)/2, every maximum intersecting subfamily of independent r-sets of G is a star, and that every such maximum family is a star when 1<r<μ(G)/2.

Background

The Holroyd–Talbot conjecture generalizes the Erdős–Ko–Rado theorem from r-subsets of an empty graph to independent r-sets of an arbitrary graph. Here μ(G) denotes the size of a smallest maximal independent set, and an r-star is a family of independent r-sets containing a common vertex. The survey presents this conjecture as the organizing problem and records numerous graph classes for which it has been verified, while the general statement remains unresolved.