Optimal index conjecture for terminal and canonical Calabi-Yau varieties

Let $n$ be an integer at least $2$. Let $X$ be the terminal Calabi-Yau $n$-fold constructed in Corollary, with index

$$m = (s_{n-1}-1)(2s_{n-1}-3).$$

Optimal index conjecture. The index $m$ of $X$ is the largest possible index among all terminal Calabi-Yau varieties of dimension $n$, and also among all canonical Calabi-Yau varieties of dimension $n$.

The construction provides terminal Calabi-Yau varieties attaining the proposed bound. The conjecture asserts that this bound is optimal in both the terminal and canonical settings.