Freeness conjecture for the equivariant cohomology of a graph

Let $\Gamma$ be a graph with axial function $\alpha$, let $H(\Gamma,\alpha)$ be its equivariant cohomology ring, and let ${\mathbb S}({\mathfrak g}^*)$ be the polynomial ring occurring as the ring of constant maps. For a generic $\xi\in{\mathfrak g}$, let $\sigma_p$ be the number of ascending edges terminating at a vertex $p$, and let $b_k(\Gamma)$ be the number of vertices with $\sigma_p=k$. Freeness conjecture. $H(\Gamma,\alpha)$ is a free ${\mathbb S}({\mathfrak g}^*)$-module with $b_k(\Gamma)$ generators of degree $k$. This is one of the paper's Morse-theoretic conjectures concerning the equivariant cohomology ring of a graph; the supplied text gives no resolution status or further supporting context.