  Generalizing a question of Mukai, we conjecture that a Fano manifold $X$ with
Picard number $\rho_X$ and pseudo-index $\iota_X$ satisfies $\rho_X (\iota_X-1)
\le \dim(X)$. We prove this inequality in several situations: $X$ is a Fano
manifold of dimension $\le 4$, $X$ is a toric Fano manifold of dimension $\le
7$ or $X$ is a toric Fano manifold of arbitrary dimension with $\iota_X \ge
\dim(X)/3+1$. Finally, we offer a new approach to the general case.
