Higher-dimensional Payne-type inequality for buckling eigenvalues

Let $\Omega$ be a bounded domain in $\mathbb R^d$ with smooth boundary. Let $\lambda_j$, $\Lambda_j$, and $\sigma_j$ denote the eigenvalues used in the paper for the Dirichlet Laplacian, Dirichlet Bilaplacian, and buckling problem, respectively. Then the higher-dimensional Payne-type conjecture.

$$\Lambda_j>\lambda_j\sigma_j$$

for all $j\in\mathbb N$.

The inequality generalizes Payne's inequality, which holds for $j=1$ in $d$ dimensions, and the paper proves the analogous statement in one dimension. The supplied text does not establish the conjecture in higher dimensions.