Trench's asymptotic absolute equidistribution conjecture for random geometric graph spectra

Let $P(\mathcal{X}_n)$ and $P(\mathcal{D}_n)$ be the matrices associated with the random geometric graph and its deterministic comparison model, and let $\lambda_i(\mathcal{X}_n)$ and $\lambda_i(\mathcal{D}_n)$ denote their $i$th largest eigenvalues. Trench's asymptotic absolute equidistribution conjecture. For any $f\in\mathcal{C}[-1,1]$,

$$\lim_{n\uparrow\infty} \frac{1}{n}\sum_{i=1}^n\left|f\left(\lambda_i\left(\mathcal{X}_n\right)\right)-f\left(\lambda_i\left(\mathcal{D}_n\right)\right)\right|^2 =0.$$

The conjecture strengthens asymptotic equidistribution by requiring the matched eigenvalues to become asymptotically indistinguishable in mean square after applying every continuous function. It is presented as being suggested by simulations and attributed to Trench; the supplied text gives no resolution.