Conjecture on Calabi-energy minimization by Bach-flat Kähler metrics

Let $(M,J)$ be a compact complex surface, and let $g$ be a Bach-flat Kähler metric, meaning a Kähler metric compatible with $J$ whose Bach tensor vanishes. Let $\mathcal{C}$ denote the Calabi energy on the space of Kähler metrics on $M$. Calabi-minimizer conjecture. The metric $g$ is an absolute minimizer of the Calabi energy $\mathcal{C}$ on the space of all $J$-compatible Kähler metrics on $M$. Bach-flat Kähler metrics are critical points of the Calabi energy, and the known examples are absolute minima; whether every such metric is an absolute minimizer remains open.