Constant scalar curvature Sasakian metrics and K-polystability conjecture

Let $(M,\xi)$ be a Sasakian manifold polarized by the Reeb vector field $\xi$, with polarized affine cone $(Y,\xi)$, where $Y=C(M)$ is the fixed complex cone. A constant scalar curvature Sasakian (cscS) structure is a Sasakian structure compatible with $\xi$ and this cone, and K-polystability is the stability condition for the polarized affine variety $(Y,\xi)$. Constant scalar curvature Sasakian K-polystability conjecture. There exists a cscS structure compatible with $\xi$ and the fixed complex cone $Y=C(M)$ if and only if $(Y,\xi)$ is K-polystable. This is the Sasakian analogue of the Yau–Tian–Donaldson conjecture. The paper notes that existence implies K-semistability, while the converse equivalence is presented as a natural conjecture and its general resolution is not supplied.