Existence of a divisor computing the global log canonical threshold of a Fano variety

Let $X$ be a Fano variety with log terminal singularities. An effective $\mathbb{Q}$-divisor $D$ on $X$ satisfies $D\equiv -K_X$.

Global threshold conjecture. For some such divisor $D$, one has

$$\mathrm{lct}\big(X\big)=\mathrm{lct}\big(X,D\big).$$

This conjecture asks whether the global log canonical threshold of a Fano variety is attained by a single effective anticanonical $\mathbb{Q}$-divisor. It is inspired by a question of Tian; the supplied source does not establish a general resolution.