Random cellular-semimetric graph Laplacian convergence on the Sierpiński gasket

Let $\mathrm{SG}$ be the Sierpiński gasket, let $\mathcal{L}$ be its Laplacian, and let $\mathcal{L}_\epsilon(\omega)$ be the random graph averaging Laplacian defined from random sample points using the cellular semimetric. Suppose $f:\mathrm{SG}\to\mathbb{R}$ has vanishing normal derivatives at $\{p_1,p_2,p_3\}$ and $\mathcal{L}f$ is Hölder continuous with exponent $\alpha$. Random graph convergence conjecture. There are $\beta>1$ and $C>0$ such that, whenever $\epsilon(n)\to0$ and $n\epsilon(n)^\beta\to\infty$, for every $x\in\mathrm{SG}\setminus\{p_1,p_2,p_3\}$,

$$\mathbb{E}\left|\epsilon^{-\frac{\log 5}{\log 2}}\mathcal{L}_\epsilon(\omega)f-c\mathcal{L}f\right|\leq Cn^{-\gamma},$$

where $\gamma=\gamma(\alpha,\beta)>0$. This is the random-sampling analogue of cellular-semimetric averaging convergence; the normal derivative condition supplies the Neumann-type boundary hypothesis.