Lu's second-gap conjecture for minimal submanifolds

Let $M^n$ be a closed minimal submanifold of the unit sphere $\mathbb S^{n+m}$. With respect to a local orthonormal normal frame, let $\mathcal A$ be the fundamental matrix of the shape operators, and let $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_m\geq0$ be its eigenvalues. Write $S=\operatorname{tr}\mathcal A$. Lu's second-gap conjecture. For every pair $(n,m)$ there exists $\varepsilon(n,m)>0$ with the following property: if $S+\lambda_2$ is constant and $S+\lambda_2>n$, then

$$S+\lambda_2>n+\varepsilon(n,m).$$

This conjecture proposes a second gap above Lu's first-gap threshold $n$ for the refined curvature quantity $S+\lambda_2$. It is motivated by the classical second-gap theorem of Peng and Terng and remains unresolved in the supplied source.