Two-term asymptotic conjecture for the buckling counting function

Let $\Omega$ be a bounded domain in $\mathbb R^d$ with smooth boundary. Let $N(z)$ denote the counting function for the eigenvalues of the buckling problem, and let $B_k$ denote the volume of the unit ball in $\mathbb R^k$. Then the two-term buckling asymptotic conjecture.

$$N(z)=(2\pi)^{-d}B_d|\Omega|z^{\frac{d}{2}} -(2\pi)^{1-d}B_{d-1}|\partial\Omega|z^{\frac{d}{2}-\frac{1}{2}}\left(\frac{1}{4}+\frac{\Gamma\left(\frac{d}{2}\right)}{2\pi^{\frac{1}{2}}\Gamma\left(\frac{d}{2}+\frac{1}{2}\right)}\right)+o\left(z^{\frac{d}{2}-\frac{1}{2}}\right).$$

Two-term asymptotic expansions are known for related Laplacian and Bilaplacian problems, but are not known here because the buckling eigenvalues arise from an operator pencil. The conjecture is based on formal considerations; its status is not established in the supplied text.