The effective anticanonical divisor conjecture for global log canonical thresholds

Let $X$ be a del Pezzo variety with log terminal singularities, and let $\mathrm{lct}(X)$ denote its global log canonical threshold. An effective $\mathbb{Q}$-divisor $D$ on $X$ is anticanonically equivalent when it is $\mathbb{Q}$-linearly equivalent to $-K_X$.

Effective anticanonical divisor conjecture. There is an effective $\mathbb{Q}$-divisor $D$ on $X$ such that

$$D\sim_{\mathbb{Q}}-K_X$$

and

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

The claim would reduce the computation of the global log canonical threshold to the log canonical threshold of a single effective anticanonical $\mathbb{Q}$-divisor. The surrounding discussion notes that rationality of global log canonical thresholds is unknown even for del Pezzo surfaces with log terminal singularities; no resolution of this conjecture is given here.