The K-energy properness criterion for constant scalar curvature Kähler metrics

Let $(M,\omega,J)$ be a compact Kähler manifold, and let $K$ be a maximal compact subgroup of $\operatorname{Aut}_0(M)$. Let ${\mathcal H}$ be the space of Kähler potentials and let ${\mathcal H}_K\subset{\mathcal H}$ be the subset of $K$-invariant metrics. The K-energy properness conjecture. The space ${\mathcal H}$ contains a constant scalar curvature metric if and only if the ${\mathcal K}$-energy is proper on ${\mathcal H}_K$. Darvas and Rubinstein disproved this statement, so it is refuted.