Donaldson's energy lower-bound conjecture for the Kähler metric space

Let $V$ be a Kähler manifold, let $\mathcal{H}$ be the space of Kähler potentials, and let $\mathcal{H}_0$ denote its normalized subspace. For a normalized $\phi\in\mathcal{H}_0$, let $\widetilde{\phi}_t$, $t\in[0,1]$, be any path from $0$ to $\phi$ in $\mathcal{H}$, and let $d\mu_{\widetilde{\phi}_t}$, $d\mu_\phi$, and $d\mu_0$ be the corresponding volume measures. Donald's energy lower-bound conjecture. There is a constant $M$ such that

$$\int_0^1\int_V\left(\frac{d\widetilde{\phi}}{dt}\right)^2d\mu_{\widetilde{\phi}_t}\,dt\geq M^{-1}\left(\max\left(\int_{\phi>0}\phi\,d\mu_\phi,-\int_{\phi<0}\phi\,d\mu_0\right)\right)^2.$$

This lower bound would imply positivity of the path-energy infimum for distinct endpoints and hence help establish that $\mathcal{H}$ is a metric space. The supplied text does not state whether this quantitative conjecture is resolved.