Donaldson's long-time asymptotics conjecture for the Calabi flow

Let $(X,\omega)$ be a compact Kähler manifold, let $\mathcal H_\omega$ denote the space of Kähler potentials in the class $[\omega]$, and let $[0,\infty)\ni t\mapsto c_t\in\mathcal H_\omega$ be a Calabi flow trajectory. A potential $c_\infty$ is csc-K if it has constant scalar curvature. Donald's conjecture. Exactly one of the following alternatives holds:

1. The curve $t\mapsto c_t$ converges smoothly to some csc-K potential $c_\infty\in\mathcal H_\omega$ as $t\to\infty$.
2. The curve $t\mapsto c_t$ diverges as $t\to\infty$ and encodes destabilizing information about the Kähler structure.

This conjecture describes the expected long-time behavior of the Calabi flow: convergence should occur when the Kähler structure admits a csc-K metric, while divergence should reflect instability. The source presents long-time existence for arbitrary initial data as an open conjecture due to Calabi and Chen, and does not provide a resolution of this Donaldson conjecture.