Convergence of random graph Laplacian eigenmaps on an interval

Let $\phi_k$ denote the $k^\text{th}$ eigenfunction of the Laplacian on $[-1,1]$, and let $\phi_{\epsilon,n,k}$ denote the $k^\text{th}$ eigenfunction of a graph Laplacian $\mathcal{L}_{\epsilon,n}$. Convergence conjecture. If $0<\beta<1$ and $\epsilon(n)\sim n^{-\beta}$, then

$$\|\phi_k-\phi_{\epsilon(n),n,k}\|_{L^\infty}\xrightarrow[]{}0$$

as $n\to\infty$. For $0<\beta<2/3$, the convergence is improved to convergence in energy, and for $0<\beta<1/3$, it is further improved to convergence of the Laplacians. These conjectured convergence rates would substantially improve the bounds supplied by the preceding theorems for low-frequency eigenfunctions.