Donaldson's relative K-stability conjecture for labeled polytopes

Let $(\Delta,u)$ be a labeled polytope. A symplectic potential is a function $G\in\mathcal{S}(\Delta,u)$, and its scalar curvature is

$$S(G)=-\sum_{ij}\frac{\operatorname{\partial}^2 H_{ij}}{\operatorname{\partial}\mu_i\operatorname{\partial}\mu_j}.$$

The labeled polytope is analytically relatively $K$-stable with respect to toric degenerations when the relative Futaki functional

$$\mathcal{L}_{\Delta,u}(f)=\int_{\operatorname{\partial}\Delta}f\,d\nu-\frac{1}{2}\int_{\Delta}f\zeta_{(\Delta,u)}\,dv$$

is non-negative on every convex continuous piecewise affine-linear function on $\Delta$, and vanishes only for affine-linear functions. Here $\zeta_{(\Delta,u)}$ is the extremal affine-linear function, and $H_{ij}$ is the inverse Hessian of $G$.

Donaldson's relative K-stability conjecture. There is a solution $G\in\mathcal{S}(\Delta,u)$ to the extremal Kähler equation

$$S(G)=\zeta_{(\Delta,u)}$$

if and only if $(\Delta,u)$ is analytically relatively $K$-stable with respect to toric degenerations.

This conjecture formulates the expected equivalence between the existence of extremal Kähler metrics on toric orbifolds and relative stability under toric degenerations. The supplied context attributes the precise conjecture to Donaldson; its general resolution status is not established in the source material.