Eigenvalue bound conjecture for covariance matrices of volume power functionals

Let $\Sigma_n$ be the covariance matrix associated with the volume power functional in the critical regime. Assume that, for every $m\in\{0,\ldots,k_n\}$, the matrices $A_m^{>1}$ satisfy Requirement 1, and let $S_m$ be the corresponding bound. Denote the eigenvalues of $\Sigma_n$ by

$$\lambda_1^{\Sigma_n}\leq\cdots\leq\lambda_n^{\Sigma_n}.$$

Eigenvalue bound conjecture. These eigenvalues satisfy

$$0\leq\lambda_1^{\Sigma_n}\leq\cdots\leq\lambda_n^{\Sigma_n}\leq (k_n+1)\max_{m=0,\ldots,k_n}S_m\left(\sum_{i=1}^n\frac{1}{a_{i,m}^2}\right).$$

The bound is proposed for the covariance matrix in the critical regime and follows the preceding spectral estimates for the matrices $A_m^{>1}$, subject to the stated requirement. Its resolution would provide a uniform control on the eigenvalues of $\Sigma_n$ in terms of the quantities $S_m$ and $a_{i,m}$; the supplied text gives no evidence that it has been proved or refuted.