The claw-free hard-core zero conjecture

Let $G$ be a finite claw-free graph, meaning that it has no induced subgraph isomorphic to $K_{1,3}$. Let $Z(w)$ be the hard-core lattice-gas partition function of $G$. Hamidoune–Stanley conjecture. All zeros of $Z(w)$ are negative real. This would extend the Heilmann–Lieb real-negative-zero theorem from line graphs to all claw-free graphs; the source presents it as a conjecture and gives no resolution.