Regularity conjecture for minimizers of the extended K-energy

Let $(X,\omega)$ be a compact Kähler manifold, let $(\mathcal E^1,d_1)$ be the metric completion of the space of Kähler potentials, and let the extended Mabuchi K-energy be a map $\mathcal K:\mathcal E^1\to(-\infty,+\infty]$. A point $u\in\mathcal E^1$ is a minimizer of $\mathcal K$ if it attains the infimum of $\mathcal K$ on $\mathcal E^1$. A Kähler metric is constant scalar curvature Kähler (cscK) if its scalar curvature is constant. Regularity conjecture. If $u$ is a minimizer of the extended K-energy, then $u$ is a smooth cscK metric. This conjecture is presented as the regularity question linked to the general modified properness conjecture; the supplied text gives no resolution.