The Fano Kähler–Ricci soliton conjecture for Sasaki–Einstein circle bundles

Let $M$ be a Fano manifold. A Kähler–Ricci soliton on $M$ is a Kähler metric satisfying the Kähler–Ricci soliton equation. Let $K_M$ be the canonical line bundle of $M$, and consider the associated $S^1$-bundle. The Reeb field is the vector field defining the corresponding Reeb foliation and its transverse geometry.

Sasaki–Einstein circle-bundle conjecture. If there exists a Kähler–Ricci soliton on $M$, then the $S^1$-bundle associated to $K_M$ admits a Sasaki–Einstein metric with a suitable choice of the Reeb field.

This conjecture relates Kähler–Ricci solitons on Fano manifolds to Sasaki–Einstein metrics on their associated circle bundles. The statement is presented as a conjecture in the source; its resolution is not specified there.