Donaldson's existence conjecture for extremal metrics on toric manifolds

Let $P\subset \mathbb{R}^{n}$ be a Delzant polytope, let $A$ be a smooth function on $\overline{P}$, and let $L_A$ be the associated linear functional on convex functions. An admissible symplectic potential is a symplectic potential on $P$ with the prescribed boundary behaviour for the Delzant polytope. Donaldson's existence conjecture. If

$$L_A(f)=0$$

for every affine-linear function $f$ and

$$L_A(f)>0$$

for every convex function $f$ that is not affine-linear, then there is an admissible symplectic potential $u$ satisfying

$$u_{ij}^{ij}=-A.$$

This is the proposed sufficiency condition for solving the Abreu equation on a toric manifold. The preceding discussion explains that failure of the condition obstructs existence and that degenerating sequences can produce piecewise-linear convex functions; the conjecture asserts that these are the only obstructions, but the supplied text does not state a resolution.