Index conjecture for constant mean curvature hypersurfaces in spheres

Let $\Sigma^n$ be a compact orientable constant mean curvature hypersurface immersed into $\mathbb{S}^{n+1}$. Its weak index, denoted by $\operatorname{Ind}_T(\Sigma)$, is the natural index for studying stability of constant mean curvature hypersurfaces.

CMC index conjecture. Either

$$\operatorname{Ind}_T(\Sigma)=0,$$

with $\Sigma$ a totally umbilical sphere in $\mathbb{S}^{n+1}$, or

$$\operatorname{Ind}_T(\Sigma)\geq n+2,$$

with equality if and only if $\Sigma$ is a constant mean curvature Clifford torus

$$\mathbb{S}^k(r)\times\mathbb{S}^{n-k}\left(\sqrt{1-r^2}\right),$$

where

$$\sqrt{\frac{k}{n+2}}\leq r\leq\sqrt{\frac{k+2}{n+2}}.$$

The conjecture extends the index-boundary phenomenon from minimal hypersurfaces to constant mean curvature hypersurfaces. Totally umbilical spheres are known to be the only compact weakly stable constant mean curvature hypersurfaces, while the supplied text does not state whether the conjectured sharp classification is resolved.