Around the Mukai conjecture for Fano manifolds

As a generalization of the Mukai conjecture, we conjecture that the Fano manifolds $X$ which satisfy the property $\rho_X(r_X-1)\geq\dim X-1$ have very special structure, where $\rho_X$ is the Picard number of $X$ and $r_X$ is the index of $X$. In this paper, we classify those $X$ if $\rho_X\leq 3$ or $\dim X\leq 5$.