Donaldson's conjecture on conical Kähler–Einstein metrics and the greatest Ricci lower bound

Let $M$ be a Fano manifold and let $D\in |-K_M|$ be a smooth divisor. Define

$$R(M)=\sup\{t\mid \exists\,\omega\in c_1(M)\text{ such that }\operatorname{Ric}(\omega)>t\omega\}$$

and let $R(M,D)$ be the supremum of the $b2>0$ for which there exists a Kähler–Einstein metric with cone angle $2\pi\beta$ along $D$ satisfying

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

Donald's conjecture. For all $0<\beta<R(M)$ there exists a cone-singularity solution to this equation, and there is no solution for $R(M)<\beta<1$. Equivalently,

$$R(M,D)=R(M)$$

for every smooth $D\in |-K_M|$. The conjecture relates the conical existence threshold to the greatest Ricci lower bound of the underlying Fano manifold. The cited results establish analogous solvability statements when the divisor current is replaced by a smooth Kähler form, while the conical case is posed here as a conjecture.