Generalized Mukai conjecture on the pseudoindex and Picard number

Let $X$ be an $n$-dimensional smooth Fano variety, and let its pseudoindex be

$$\iota_X:=\min\{-K_X\cdot C\mid C\subset X\text{ is a rational curve}\}.$$

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

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

with equality if and only if $X$ is isomorphic to $({\mathbb P}^{\iota_X-1})^{\rho_X}$. This replaces the index by the pseudoindex and extends Mukai's conjecture; the claim is not resolved in general, although several cases are known.