The modified uniform K-stability conjecture for toric extremal metrics

Let $(M,\omega)$ be a toric manifold and let $\Delta$ be the corresponding Delzant polytope. Let $A$ be the function appearing in the Abreu equation, and let uniformly K-stable mean uniform K-stability of the toric data $(\Delta,A)$. A metric within the class $[\omega]$ is said to solve the Abreu equation when it satisfies the equation governing toric extremal metrics.

Modified uniform K-stability conjecture. The pair $(M,\omega)$ has a metric within the class $[\omega]$ that solves the Abreu equation if and only if $(\Delta,A)$ is uniformly K-stable. In particular, the metric is extremal if and only if $A$ is affine linear.

The paper proposes this as a modification of Donaldson's conjecture. It proves that uniform K-stability is necessary for the existence of extremal metrics on toric varieties, and notes that uniform K-stability implies existence of a weak solution; the full equivalence is not established in the supplied text.