Tian's conjecture on conical Kähler–Einstein metrics

Let $(M,J)$ be a closed Kähler manifold, and let $D=D_1+\begingroup D_2+\cdots+D_r\endgroup\subset M$ be a divisor with simple normal crossing support. Let $[D_j]$ denote the divisor class, and let $0<\beta_j\leq 1$ be the cone-angle parameters. Suppose that

$$c_1(M,J)-\sum_{i=1}^r(1-\beta_j)[D_j]$$

is negative or zero. A conical Kähler–Einstein metric is a Kähler–Einstein metric with cone angle $2\pi\beta_j$ along $D_j$.

Tian's conjecture. Then $(M,J)$ admits a conical Kähler–Einstein metric with angle $2\pi\beta_j$ along $D_j$, unique in the negative case and unique in each Kähler class in the zero case.

The source presents this as Tian's 1994 generalization of Calabi's conjecture, but the supplied text gives no resolution status.