The pinching gap conjecture for minimal submanifolds in spheres

Let $M$ be an $n$-dimensional minimal submanifold in $S^{n+m}$. Let $\lambda_2$ denote the quantity used in the paper, and suppose that $||\sigma||^2+\lambda_2$ is constant. Pinching gap conjecture. If

$$||\sigma||^2+\lambda_2>n,$$

then there is a constant $\varepsilon(n,m)>0$ such that

$$||\sigma||^2+\lambda_2>n+\varepsilon(n,m).$$

This conjecture proposes a gap above the pinching value $n$ for the quantity $||\sigma||^2+\lambda_2$. The supplied text gives no resolution, so its status remains open.