The toric Sasaki polystability conjecture

Let $M$ be a compact co-oriented toric contact manifold of Reeb type. Let $\mathcal{C}$ be its strictly convex good cone, let $\mathcal{C}_+^*$ denote the Reeb cone, and for $b\in\mathcal{C}_+^*$ let $(\Delta_b,u_b)$ be the associated labeled polytope and $\mathcal{F}_b\in(\mathbb R^{n+1}/\mathbb Rb)^*$ the Futaki–Sasaki invariant restricted to the torus Lie algebra. Toric Sasaki polystability conjecture. The manifold $M$ admits a compatible toric constant scalar curvature Sasaki metric if and only if there exists $b\in\mathcal{C}_+^*$ such that

$$\mathcal{F}_b=0$$

and $(\Delta_b,u_b)$ is polystable.

This gives the Donaldson–Tian–Yau conjecture a toric Sasaki interpretation: vanishing of the Futaki–Sasaki obstruction together with polystability of the associated labeled polytope should characterize existence of compatible toric cscS metrics. For a fixed Reeb vector, uniqueness follows from uniqueness for the extremal Kähler equation, while the existence criterion stated here is the unresolved part.