Uniform spectral convergence of Euclidean graph approximations on the Sierpiński gasket

Let $\mathrm{SG}$ be the Sierpiński gasket with Laplacian $\mathcal{L}$. Consider the rescaled averaging Laplacian and the rescaled graph Laplacian at random sample points, both defined using the Euclidean metric on $\mathrm{SG}$. Euclidean spectral convergence conjecture. With a proper choice of normalization and alignment, their eigenvalues and eigenfunctions converge uniformly to those of $\mathcal{L}$ as $n\to\infty$ and $\epsilon(n)\to0$. The conjecture leaves the optimal behavior of $\epsilon(n)$ for the most accurate convergence unknown and is motivated by numerical investigations and comparison with the cellular semimetric.