Donaldson's geodesic stability conjecture for constant scalar curvature metrics

Let $(M,[\omega_0])$ be a polarized Kähler manifold, let ${\mathcal H}$ denote its space of Kähler potentials, and let ${\mathcal K}$ be Mabuchi's K-energy. A geodesic ray is a path $\phi(t)$ in ${\mathcal H}$ along which the geodesic equation holds.

Donald's geodesic stability conjecture. The following are equivalent:

1. There is no constant scalar curvature metric in $(M,[\omega_0])$.
2. There is a geodesic ray $\phi(t)$, $t\in[0,1)$, such that

$$\frac{d{\mathcal K}}{dt}=-\int_M\dot\phi(R-\underline{R})\omega_\phi^n<0$$

for all $t\in[0,\infty)$. 3. For any Kähler potential $\phi$, there exists a geodesic ray as in (2) starting at $\phi$.

The conjecture remains open in both directions. The source notes that insufficient regularity of geodesic rays creates a technical difficulty in interpreting the derivative of Mabuchi's K-energy.