The index-one Clifford hypersurface classification conjecture for projective spaces

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ and let $\mathbb{K}P^{r-1}$ be the corresponding projective space. A minimal Clifford hypersurface is a Clifford hypersurface that is minimal in $\mathbb{K}P^{r-1}$; a compact minimal hypersurface has index one when its Morse index is one.

Index-one Clifford hypersurface classification conjecture. The only compact minimal hypersurfaces with index one in $\mathbb{K}P^{r-1}$ are the minimal Clifford hypersurfaces. In particular, they can be minimal geodesic spheres.

This is proposed as a generalization of the paper's real-projective-space result and follows the preceding theorem asserting index one for Clifford hypersurfaces in the real and complex cases. The supplied text gives no resolution of the generalization, so it remains open.