Thom-class conjecture for the equivariant cohomology of a graph

Let $\Gamma$ be a graph with axial function $\alpha$. For each vertex $p$, let $\sigma_p$ be its index, let $F_p$ be the set of vertices that can be joined to $p$ by an ascending path, and let $E_p^-$ be the set of descending edges in $E_p$. Let $H(\Gamma,\alpha)$ be the equivariant cohomology ring and let $\mathbb S({\mathfrak g}^*)$ be the polynomial ring of constant maps. Thom-class conjecture. $H(\Gamma,\alpha)$ is freely generated as an $\mathbb S({\mathfrak g}^*)$-module by a family of Thom classes

$$\tau_p^+\in H^k(\Gamma,\alpha),\qquad k=\sigma_p,$$

satisfying

$${\,}{support}\tau_p^+\subseteq F_p$$

and

$$\tau_p^+(p)=\prod_{e\in E_p^-}\alpha_e\quad(=:\nu_p^+).$$

This gives a more explicit Morse-theoretic prediction than freeness alone, specifying generators, their degrees, supports, and values at their associated vertices; the supplied text gives no resolution status.