Second pinching conjecture for closed minimal hypersurfaces

Let $M^n$ with $n>3$ be a closed minimal hypersurface in $\mathbb{S}^{n+1}(1)$ with constant scalar curvature. Write $S$ for the squared norm of its second fundamental form.

Second pinching conjecture. If $S>n$, then $S\geq 2n$.

This conjecture asks whether $2n$ is the next possible pinching threshold after the known values $0$ and $n$, motivated by Cartan's isoparametric examples. Results under additional constancy assumptions on higher symmetric functions give weaker bounds, while the full assertion remains open.