Mukai's generalized conjecture for Fano manifolds

Let $X$ be a Fano manifold, meaning a smooth complex projective variety with ample anti-canonical line bundle. Let $n$ be its dimension, $\rho_X$ its Picard number, and $i_X$ its pseudo-index.

Bonavero–Casagrande–Debarre–Druel conjecture. One has

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

with equality if and only if

$$X\cong (\mathbb{P}^{i_X-1})^{\rho_X}.$$

This generalizes Mukai's conjecture, replacing the index by the pseudo-index. The conjecture is known in several cases, and this paper proves it for Fano sixfolds admitting a contraction of fiber type or having no small contractions; the full statement remains unresolved.