On the Picard number of Fano 6-folds with a non-small contraction

A generalization of S. Mukai's conjecture says that if $X$ is a Fano $n$-fold with Picard number $\rho_X$ and pseudo-index $i_X$, then $\rho_X(i_X-1) \leq n$, with equality if and only if $X \cong (\mathbb{P}^{i_X-1})^{\rho_X}$. In this paper, we prove that this conjecture holds if $n=6$ and either $X$ admits a contraction of fiber type or $X$ admits no small contractions.