Chen's regularity conjecture for K-energy minimizers

Let $(M,[\omega],J)$ be a compact Kähler manifold, let

$$\mathcal H_\omega=\{\phi\in C^\infty(M):\omega_\phi=\omega+\sqrt{-1}\partial\bar\partial\phi>0\},$$

and let $\mathcal K$ be the K-energy on Kähler potentials. A $C^{1,\bar 1}$ potential is a weak Kähler potential whose complex Hessian is bounded, and a minimizer of $\mathcal K$ is one attaining the minimum in the given Kähler class. Chen's conjecture. A $C^{1,\bar 1}$ minimizer in a given Kähler class of the K-energy is a smooth Kähler metric with constant scalar curvature. The cited work proves this regularity statement in part by showing that weak CSCK solutions with a uniform $L^\infty$ bound are smooth; the supplied evidence indicates that the broader conjecture is regarded as resolved through the known minimizer and convexity results.