Minimal two-spheres of low index in manifolds of positive complex sectional curvature

Suppose that $S^n$ is given a Riemannian metric of positive complex sectional curvatures for which the minimal two-spheres of Morse index $\leq 2n-5$ are all Morse nondegenerate. Then the number of minimal two spheres of Morse index $k$, for $n-2 \leq k \leq 2n-5$, is at least $p_{3}(k-n+2)$, where $p_{3}(k-n+2)$ is the number of $(k-n+2)$-cells in the Schubert cell decomposition for $G_3({\mathbb R}^{n+1})$.