Mukai's conjecture on the index and Picard number of Fano varieties

Let $X$ be an $n$-dimensional smooth Fano variety, with index

$$i_X:=\max\{m\in{\mathbb Z}_{>0}\mid -K_X=mL\text{ for some }L\in\operatorname{Pic}(X)\}.$$

Let $\rho_X$ be the Picard number of $X$. Mukai's conjecture. One has

$$\rho_X(i_X-1)\leq n,$$

with equality if and only if $X$ is isomorphic to $({\mathbb P}^{i_X-1})^{\rho_X}$. This conjecture relates the index and Picard number of a Fano variety; its specific case asserting that $2i_X\geq n+2$ implies $\rho_X=1$ unless $X\cong({\mathbb P}^{i_X-1})^2$ was proved by Wiśniewski, while the full statement remains open in general.