Donaldson's destabilizing geodesic ray conjecture for cscK metrics

Let $(M,[\omega])$ be a polarized Kähler manifold, let $\mathcal H$ be the space of Kähler potentials in the class $[\omega]$, and let $\mathcal H_0$ denote its normalized subspace. Assume $Aut_0(M,J)=0$.

Donaldson's conjecture. The following statements are equivalent: (1) there is no constant scalar curvature Kähler metric in $\mathcal H$; (2) there is a potential $\varphi_0\in\mathcal H_0$ and a geodesic ray $\rho(t)$, $t\in[0,\infty)$, in $\mathcal H_0$ initiating from $\varphi_0$ such that the $K$-energy is non-increasing; (3) for every Kähler potential $\psi\in\mathcal H_0$, there is a geodesic ray $\rho(t)$, $t\in[0,\infty)$, in $\mathcal H_0$ initiating from $\psi$ such that the $K$-energy is non-increasing.

This conjecture relates nonexistence of cscK metrics to destabilizing geodesic rays in the space of Kähler potentials. The paper states that it is proved under the assumption of discrete automorphism group.