Rayleigh conjecture for graph component-size weights

Let $\mathsf G=(V,E)$ be a finite connected graph. For each independent edge set $S$ of $\mathsf G$, let $\omega_G(S)$ be the product of the sizes of the connected components of $(V,S)$, and let $\omega_G$ vanish outside the independent sets. Graph component-weight Rayleigh conjecture. The weight function $\omega_G$ is Rayleigh. The source gives real-rootedness of an associated generating polynomial but leaves this stronger Rayleigh assertion open.