Weak but not strong Euclidean averaging convergence on the Sierpiński gasket

Let $\mathrm{SG}$ be the Sierpiński gasket with Laplacian $\mathcal{L}$, and suppose $\mathcal{L}f$ is Hölder continuous on $\mathrm{SG}$. Let $\mathcal{L}_\epsilon$ and $\mathcal{L}_{\epsilon,n}(\omega)$ be the averaging and random graph Laplacians defined using the Euclidean metric. Weak-convergence and strong-obstruction conjecture. There is a sequence $\epsilon_j\to0$ such that

$$\epsilon_j^{-\frac{\log 5}{\log 2}}\mathcal{L}_{\epsilon_j}(\omega)f\to c\mathcal{L}f$$

as $j\to\infty$ in the weak-$L^2$ topology, and similarly for $\mathbb{E}\mathcal{L}_{\epsilon,n}(\omega)$, but this convergence does not hold in the strong-$L^2$ topology for any sequence $\epsilon_j$. This predicts a sharp distinction between weak and strong Euclidean averaging convergence on the gasket.