Bonavero–Casagrande–Debarre–Druel generalized Mukai conjecture

Let $X$ be a smooth Fano variety. Let $\rho_X$ denote its Picard number and let $\iota_X$ denote its pseudo-index, defined by

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

Here $-K_X$ is the anticanonical divisor of $X$. The generalized Mukai conjecture. One has

$$\rho_X(\iota_X-1)\leq\dim X,$$

and equality holds if and only if $X$ is isomorphic to $(\mathbb P^{\iota_X-1})^{\rho_X}$. This conjecture has been proved in many cases, including the setting of the paper for spherical varieties with reductive general isotropy group; the general statement is not asserted here to be resolved.