Jiao's bounded complement conjecture for Calabi–Yau type varieties

Let $X$ be a $d$-dimensional $\varepsilon$-lc Calabi--Yau type variety, and let $A$ be an ample divisor on $X$ such that $A^d\leq v$, where $\varepsilon,d,v>0$.

Jiao's conjecture. There exists a positive integer $m$ with the following property: there is an effective $\mathbb{Q}$-Weil divisor $\Delta$ on $X$ such that $(X,\Delta)$ is klt and

$$\nm(K_X+\Delta)\sim 0.$$

This predicts a uniform Cartier index under the stated boundedness assumption for $\varepsilon$-lc Calabi--Yau type varieties. The source gives no resolution status, so the conjecture remains open.