The sharp Morrey embedding conjecture for Sierpinski carpet

Let $d_H$ be the Hausdorff dimension, $d_W$ the walk dimension, $d_{tH}$ the topological Hausdorff dimension, and let $\delta_{\mathcal{E}}$ be the critical Sobolev-embedding exponent for the Dirichlet form $\mathcal{E}$. For $f\in W^{1,p}(\mathcal{E})$, set

$$\lambda=\frac{(d_W-d_H+d_{tH}-1)(p-2)+d_W}{p}-\frac{d_H}{p}.$$

Morrey embedding conjecture. For the Sierpinski carpet,

$$\delta_{\mathcal{E}}=2-\frac{d_W-d_H}{d_W-d_H+d_{tH}-1},$$

and, for every $p>\delta_{\mathcal{E}}$, there exists $C>0$ such that

$$\mu\text{-ess}\sup\limits_{x\neq y}\frac{|f(x)-f(y)|}{d(x,y)^\lambda}\leq C\mathbf{Var}_{p,\mathcal{E}}(f)$$

for every $f\in W^{1,p}(\mathcal{E})$. This is the Morrey-embedding consequence of the conjectured Besov exponents for the carpet and remains open in the supplied text.