The generalised Mukai conjecture for smooth Fano varieties

Let $X$ be a smooth Fano variety. Denote by $\rho_X$ its Picard number, by $\iota_X$ its pseudo-index, and by $\dim X$ its dimension. The generalised Mukai conjecture.

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

with equality if and only if

$$X\cong {\left(\mathbb{P}^{\iota_X-1}\right)}^{\rho_X}.$$

This conjecture generalises Mukai's characterisation of products of projective spaces among smooth Fano varieties. It has been verified in various special cases, but remains open in general.