Convexity package for momentum maps on contact manifolds

Let a torus T act effectively on a compact connected cooriented contact manifold, and let Ψ be the natural momentum map on the symplectization. We prove that, if dim T > 2, the union of the origin with the image of Ψ is a convex polyhedral cone, the nonzero level sets of Ψ are connected (while the zero level set can be disconnected), and the momentum map is open as a map to its image. This answers a question posed by Eugene Lerman, who proved similar results when the zero level set is empty. We also analyze examples with dim T ≤ 2.