Donaldson's K-polystability conjecture for conformally Kähler Einstein-Maxwell metrics

Let $(\Delta, {\bf L})$ be a labelled Delzant polytope and $f$ a positive affine function on $\Delta$. A solution of the modified Abreu equation belongs to the space $\mathcal{S}(\Delta, {\bf L})$, and $(\Delta, {\bf L}, f)$ is K-polystable when the Donaldson–Futaki invariant satisfies $\mathfrak{F}_{\Delta, {\bf L}, f}(\varphi)\geq 0$ for every piecewise affine linear convex function $\varphi$ on $\Delta$, with equality only when $\varphi$ is affine linear.

Donaldson's conjecture. There exists a solution of the modified Abreu equation in $\mathcal{S}(\Delta, {\bf L})$ if and only if $(\Delta, {\bf L}, f)$ is K-polystable.

This is the main conjecture in the theory relating conformally Kähler Einstein–Maxwell metrics to an algebro-geometric stability condition. The supplied text gives no evidence resolving it.