Donaldson-type convexity criterion for extremal metrics on compatible classes

Let $M$ be a manifold with a compatible Kähler class $\Omega$, moment polytope $\Delta$, and associated linear functional $\mathcal F^\Omega$ on functions on $\Delta$. A compatible extremal Kähler metric means that the extremal equation has a solution in the space $\mathcal S(\Delta)$ of compatible symplectic potentials.

Convexity criterion conjecture. The following conditions should be equivalent: $\Omega$ admits an extremal Kähler metric; $\Omega$ admits a compatible extremal Kähler metric; and

$$\mathcal F^\Omega(f)\geq 0$$

for every piecewise linear convex function $f$ on $\Delta$, with equality if and only if $f$ is affine.

This recasts leading conjectures in the toric setting for rigid semisimple toric bundles. The source notes that positivity on a larger space of convex functions may generally be needed, while piecewise linear convex functions suffice in the cited special case where $\ell=2$ and the base is a point.