The Cantor graph equality conjecture for complex dimensions

Let $A$ be the Cantor graph, let $A_{1/3}$ be its Euclidean $1/3$-neighborhood, and let $\mathcal{D}_{A,A_{1/3}}$, $\mathcal{D}_{A,\Omega}$, and $\mathcal{D}_A$ denote the corresponding multisets of complex dimensions. Set $D_{CS}=\log_3 2$ and $\mathbf{p}=2pi/\log 3$.

Cantor graph complex-dimension conjecture. One should have

$$\mathcal{D}_A=\mathcal{D}_{A,A_{1/3}}=\mathcal{D}_{A,\Omega}=\{0,1\}\cup(D_{CS}+i\mathbf{p}\mathbb{Z}).$$

The conjecture strengthens the currently established inclusion for the Euclidean neighborhood and matches the dimensions predicted from the Cantor graph and Cantor string. Numerical evidence supports the presence of several nonreal conjugate dimensions, but the full equality remains unproved.