Symmetry-invariant random rotations of Sierpiński-gasket eigenmaps

Let $\mathcal{L}$ be the Laplacian on the Sierpiński gasket, whose relevant Neumann eigenspace has multiplicity two, and consider eigenmaps arising from random graph approximations with $n\to\infty$ and $\epsilon(n)\to0$ sufficiently slowly. Random-rotation conjecture. These eigenmaps produce images of nonrandom eigenmaps randomly rotated in $\mathbb{R}^2$, and the random rotations have a limiting density invariant under the symmetry group of the Sierpiński gasket. The conjecture is motivated by numerical evidence and by the two-dimensionality of the relevant Neumann eigenspace; the precise rate at which $\epsilon(n)$ should tend to zero is not specified.