Chen's geodesic properness conjecture for the K-energy

Let $(M,[\omega])$ be a Kähler manifold with $\mathcal H$ its space of Kähler potentials, and let $d$ be the geodesic distance on $\mathcal H$. Assume $Aut_0(M,J)=0$. The $K$-energy is proper in terms of geodesic distance when it tends to $+\infty$ whenever $d$ tends to infinity.

Chen's properness conjecture. A constant scalar curvature Kähler metric exists if and only if the $K$-energy is proper with respect to geodesic distance.

The original formulation used Donaldson's $L^2$ distance. The paper explains that later work identifies the $L^1$ geodesic distance as the natural version and states that this $L^1$ formulation is proved there.