Smooth maximal-index conjecture for Calabi-Yau varieties

Let $s_0=2$ and $s_{r+1}=s_0\cdots s_r+1$ be Sylvester's sequence. A smooth, projective Calabi-Yau $n$-fold is a smooth projective variety with $K_X\sim_{\mathbb Q}0$, and its index is the smallest positive integer $m$ such that $mK_X\sim0$.

Smooth maximal-index conjecture. The largest possible index of any smooth, projective Calabi-Yau $n$-fold is

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

The constructed varieties attain this value, so the conjecture would identify their index as maximal. It is a smooth-case consequence of the analogous conjecture for terminal Calabi-Yau varieties and remains open in general.