The K-stability conjecture for constant scalar curvature and extremal metrics

Let $M$ be a polarized Kähler manifold. Stability here means stability in a suitable sense of geometric invariant theory, while an extremal metric is a Kähler metric whose scalar curvature has the extremal property; constant scalar curvature metrics are the special case in which the scalar curvature is constant.

K-stability conjecture. $M$ admits a Kähler metric with constant scalar curvature, or more generally an extremal metric, if and only if $M$ is stable in a certain sense of geometric invariant theory.

This is the conjectural link between canonical Kähler metrics and algebro-geometric stability. The source presents it as a well-known conjecture and does not state a resolution, so its status here is open.