Realization conjecture for colored regular graphs

Let $(\Gamma,\alpha)$ be a finite connected regular $(n+1)$-valent graph whose edges are colored by $(\mathbb Z_2)^{n+1}$, with $\alpha$ a “good” coloring. A moment graph is the graph associated with the fixed-point data of a group action. Realization conjecture. Every such $(\Gamma,\alpha)$ can be realized as the moment graph of some $(\mathbb Z_2)^{n+1}$-action on an $(n+1)$-dimensional closed manifold. The preceding realization theorem proves this when $(\Gamma,\alpha)$ admits an $n$-skeletal expansion, while the conjecture concerns the general case.