On Chern's conjecture for minimal submanifolds of the sphere

A well-known conjecture of Chern, do Carmo, and Kobayashi asserts that, for $n,m\ge 1$, the scalar curvature of a closed, minimally immersed $n$-submanifold of ${\mathbb{S}}^{n+m}$ with second fundamental form of constant length takes values in a discrete set. This property holds in every codimension when $n\in \{1,2\}$. We disprove this conjecture for all $n\ge 3$ with $m\ge 4$, and for even $n\ge 4$ with $m\ge 3$.