Cellular-semimetric averaging conjecture on the Sierpiński gasket

Let $\mathrm{SG}$ be the Sierpiński gasket with its standard Laplacian $\mathcal{L}$, and define the cellular semimetric by

$$d_{\mathrm{cell}}(x,y):=\min\{2^{-m}:x\text{ and }y\text{ are in intersecting }m\text{-cells}\}.$$

Let $\mathcal{L}_\epsilon$ be the averaging operator over cellular-semimetric balls. Cellular-semimetric convergence conjecture. There is a constant $c$ such that

$$\epsilon^{-\frac{\log 5}{\log 2}}\mathcal{L}_\epsilon f\xrightarrow[\epsilon\to0]{}c\mathcal{L}f$$

uniformly for every function $f$ in the Hölder domain of the Laplacian. Moreover, the eigenvalues and eigenfunctions of $\mathcal{L}_\epsilon$ converge to those of $\mathcal{L}$ as $\epsilon\to0$. This conjecture is motivated by existing results on obtaining the Sierpiński-gasket Laplacian by rescaled cell averages and may be approachable using Dirichlet-form methods.