Bounded index conjecture for rationally connected klt Calabi–Yau varieties

Let $d$ be a positive integer. Consider any $d$-dimensional klt Calabi–Yau rationally connected projective variety $X$. Rationally connected Calabi–Yau index conjecture. There exists a positive integer $n$, depending only on $d$, such that

$$nK_X\sim0.$$

This is presented as a conjecture and is the subject of a later section; the supplied text gives no general resolution.