Conjecture on the diffusion and analytic structure of the Strichartz hexacarpet

Let $K$ be the Strichartz hexacarpet obtained as the limit of repeated barycentric subdivisions of a triangle. Let $\mathcal E$ be a self-similar local regular conservative Dirichlet form on $K$, with resistance scaling factor $\rho\approx1.304$ and Laplacian scaling factor $\tau=6\rho$.

Hexacarpet diffusion conjecture. The following statements hold:

1. On $K$ there exists a unique such Dirichlet form $\mathcal E$, and it is a resistance form in the sense of Kigami.
2. Simple random walks on the repeated barycentric subdivisions of a triangle, with time renormalized by $\tau^n$, converge to the diffusion process corresponding to $\mathcal E$.
3. This diffusion satisfies sub-Gaussian heat-kernel estimates and elliptic and parabolic Harnack inequalities, possibly with logarithmic corrections, for Hausdorff dimension $\dfrac{\log(6)}{\log(2)}\approx2.58$ and spectral dimension $2\dfrac{\log(6)}{\log(\tau)}\approx1.74$.
4. The spectrum of the Laplacian has spectral gaps in the sense of Strichartz.
5. The spectral zeta function has a meromorphic continuation to $\mathbb C$.

These claims would establish a detailed analytic description of diffusion on the hexacarpet, linking random walks on barycentric subdivisions with Dirichlet forms, heat-kernel behavior, spectral gaps, and spectral zeta functions. The paper presents them as consequences suggested by theoretical and numerical evidence; they remain conjectural.