Perdomo's conjecture on the total curvature of minimal hypersurfaces

Let $M$ be a closed embedded non-totally geodesic minimal hypersurface in $\mathbb S^{n+1}$, and let

$$\sigma=\dfrac{\int_M \left\lvert A\right\rvert^2}{\left\lvert M\right\rvert}.$$

Perdomo's conjecture. One has $\sigma\ge n$. Moreover, $\sigma=n$ if and only if $\left\lvert A\right\rvert^2\equiv n$ and $M$ is a Clifford torus.

This conjecture extends the corresponding two-dimensional characterization of the Clifford torus to higher-dimensional closed embedded minimal hypersurfaces. The paper presents results for several dimensions and for hypersurfaces with two distinct principal curvatures, while the full assertion is not resolved in the supplied source.