Kawamata's adjoint-ideal conjecture for Gorenstein section rings

Let $S$ be a section ring with Gorenstein rational singularities. For $N\gg 0$, set $I=S_{\geq N}$, and let $\operatorname{adj}(I^d)$ denote the adjoint, or multiplier, ideal of $I^d$.

Kawamata's conjecture in the Fano case. One has

$$\operatorname{gradedcore}(I)=\operatorname{adj}(I^d).$$

This is identified as the special case in which $X$ is Fano and $D=-K_X$, so that the preceding conjecture collapses to this equality. The source gives no resolution status.