Tian's modified properness conjecture for cscK metrics

Let $(X,\omega)$ be a Kähler manifold, let $\mathcal H$ be the normalized space of Kähler potentials cohomologous to $\omega$, let $G=\operatorname{Aut}_0(X,J)$ act on $\mathcal H$, and let $\mathcal K$ and $J_\omega$ denote the Mabuchi K-energy and Aubin's nonlinear energy functional. A Kähler metric is constant scalar curvature Kähler (cscK) if its scalar curvature is constant. Tian's modified properness conjecture. There exists a cscK metric cohomologous to $\omega$ if and only if there are constants $C,D>0$ such that

$$\mathcal K(u)\geq C\inf_{g\in G}J_\omega(g.u)-D,\qquad u\in\mathcal H.$$

The conjecture reduces to Tian's original prediction when $G$ is trivial and was proved for general Fano manifolds in the cited context; its general Kähler case is the subject of the paper's regularity approach.