Inverse-logarithmic asymptotic conjecture for the spectral error

Let $\kappa$ parametrize the operators $D_{\log}^{(\kappa,\kappa)}$, and let $\mathcal{E}(\kappa)$ denote the numerical error between their eigenvalues and the imaginary parts of the Riemann zeta zeros. Inverse-logarithmic asymptotic conjecture. The limit

$$\lim_{\kappa \to \infty} \mathcal{E}(\kappa)\ln \kappa$$

exists. Numerical comparisons with the first $1000$ zeros of $\zeta\left(\frac{1}{2}+is\right)$ suggest this inverse-logarithmic behavior; the corresponding inverse-logarithmic nature of $\epsilon(\lambda,N)$ is established in the paper, but the asserted limit for $\mathcal{E}(\kappa)$ is not proved here.