Energy-independence conjecture for limit points of rescaled eigenvalue measures

Let $d\geq 4$ and let $\mu_{L,E}^0$ be the measures associated with the finite-volume model at energy $E\in(-2d,2d)$. Let $n_d$ denote the density of states of the $d$-dimensional Laplacian $\Delta$, and let $\delta_a$ denote the point mass at $a$. Energy-independence conjecture. The limit points of $\mu_{L,E}^0$ are independent of $E\in(-2d,2d)$ and are given by

$$\sum_{k\in\mathbb Z}\int \sin(\theta)n_{d-1}(E-2\cos(\theta))\,\delta_{\pi k\sin(\theta)}\,d\theta.$$

The claim predicts an explicit universal description of the limiting measures in dimensions at least four, extending the preceding result establishing the existence of non-trivial limit points near the spectral edges. The statement is presented as a conjecture, and no resolution is supplied in the source.