Chern's discreteness conjecture for minimal submanifolds with constant scalar curvature

Let $M^n$ be a closed, minimally immersed submanifold in the unit sphere $\mathbb{S}^{n+m}$ with constant scalar curvature, equivalently with constant length of the second fundamental form, whose squared norm is denoted by $S$. Chern's conjecture. For each $n$, the set of all possible values for $S$ is discrete. The conjecture predicts a strong rigidity phenomenon for minimal submanifolds with constant scalar curvature. It is supported by known pinching and classification results, but remains open in general.