Closed-spectrum eigenvalue inequality conjecture for minimal submanifolds of spheres

Let $M^n$ be an $n$-dimensional compact minimal submanifold of the unit sphere $S^{n+p}(1)$, and let $\overline{\Lambda}_j$ denote the eigenvalues of the closed Laplace–Beltrami problem on $M^n$. Closed-spectrum eigenvalue inequality conjecture. For the relevant positive integer $j$,

$$\sum_{k=1}^{n}\overline{\Lambda}_{j+k} \leq (n+3)\overline{\Lambda}_{j} +\frac{\overline{\Lambda}_{j}^{2}}{\overline{\Lambda}_{j+1}}+n^{2}.$$

The inequality is proposed as a universal estimate for closed eigenvalues on minimal submanifolds. The supplied statement does not specify the range of $j$ or provide resolution evidence.