On Fano manifolds with an unsplit dominating family of rational curves

We study Fano manifolds $X$ admitting an unsplit dominating family of rational curves and we prove that the Generalized Mukai Conjecture holds if $X$ has pseudoindex $i_X=(\dim X)\mathrm{/}3$ or dimension $\dim X=6$. We also show that this conjecture is true for all Fano manifolds with $i_X>(\dim X)\mathrm{/}3$.