The cone-angle existence conjecture for Fano manifolds

Let $X$ be a Fano manifold, let $D$ be a smooth divisor in $-K_X$, and let $R(X)$ be the supremum of the parameters $\beta\leq 1$ for which the continuity-method equation

$$\operatorname{Ric}(\omega)=\beta\omega+(1-\beta)\rho$$

has a solution, equivalently the supremum defined by Szekelyhidi's invariant. A cone-singularity solution satisfies

$$\operatorname{Ric}(\omega)=\beta\omega+2\pi(1-\beta)[D].$$

The cone-angle existence conjecture. There is a cone-singularity solution $\omega_\beta$ to (27) for every parameter $\beta\in(0,R(X))$. If $R(X)<1$, there is no solution for parameters $\beta\in(R(X),1)$. This conjecture predicts that the invariant controlling the ordinary continuity method also gives the precise threshold for existence of Kähler–Einstein metrics with cone singularities. Existence for sufficiently small positive cone angles is supported by results of Tian and Yau and by the corresponding continuity-method theory, while the asserted sharp threshold and nonexistence above it remain open in the source.