Donald's uniqueness and variational conjecture for the homogeneous complex Monge–Ampère equation

Let $V$ be the compact complex manifold underlying the forms in the statement, let $R$ be a compact Riemann surface with boundary, and let $\rho:V\times\partial R\to\mathbf{R}$ satisfy that $\omega_0-\sqrt{-1}\,\overline{\partial}\partial\rho$ is a strictly positive $(1,1)$-form on each slice $V\times\{z\}$ for every $z\in\partial R$. Let $\mathcal{S}_\rho$ be the set of functions $\Phi$ on $V\times R$ equal to $\rho$ on the boundary and such that $\omega_0-\sqrt{-1}\,\overline{\partial}\partial\Phi$ is strictly positive on every slice $V\times\{w\}$, $w\in R$. Donald's conjecture. There is a unique solution $\Phi\in\mathcal{S}_\rho$ of the homogeneous complex Monge–Ampère equation

$$(\Omega_0-\sqrt{-1}\,\overline{\partial}\partial\Phi)^{n+1}=0,$$

and this solution realizes the absolute minimum of the functional $I_\rho$. This conjecture concerns the Dirichlet problem governing geodesics in the space of Kähler metrics; the supplied text gives no evidence that the asserted existence, uniqueness, or minimizing property has been resolved.