The conjecture on complex dimensions of Cartesian products

Let $A_1$ and $A_2$ be bounded subsets of $\mathbb{R}^{N_1}$ and $\mathbb{R}^{N_2}$, respectively, with $coverline{\dim}_B A_j<N_j$ for $j=1,2$. Let $\mathfrak{D}(A_j)$ be the divisor of the fractal zeta function $\zeta_{A_j}$, and let $\mathcal{D}(A_j)$ be its multiset of complex dimensions. For multisets, let $E_1+E_2$ denote the Minkowski sum.

Complex dimensions of Cartesian products. One should have

$$\mathfrak{D}(A_1\times A_2)=\mathfrak{D}(A_1)+\mathfrak{D}(A_2),$$

along with

$$\mathcal{D}(A_1\times A_2)\subseteq\mathcal{D}(A_1)+\mathcal{D}(A_2),$$

and, typically or generically, equality in the latter formula.

The divisor identity reflects the expected product structure of fractal zeta functions, while the inclusion allows zero-pole cancellations; generically, such cancellations should not occur. The analogous statement is expected for relative fractal drums.