Periodic replication conjecture for zero-dipole configurations

Let $\Omega$ be the periodic domain, let $(\widetilde U,\widetilde V)$ satisfy $|\widetilde U|=m$ and $|\widetilde V|=\zeta m$, and let the configuration have zero dipole moment

$$\int_\Omega \vec x\left(\bm1_{\widetilde U}(\vec x)-\frac{\bm1_{\widetilde V}(\vec x)}{\zeta}\right)\,\mathrm d\vec x=\vec0.$$

For each positive integer $j$, let $(\widetilde U_j,\widetilde V_j)$ be the juxtaposition of $j^n$ copies, with $j$ copies along each coordinate direction, and let $E$ and $\widetilde E$ denote the finite-periodic and infinite-periodic energies, respectively.

Periodic replication conjecture. Without requiring $(\widetilde U,\widetilde V)$ to be a global minimizer or compactly supported in $\Omega$, one has

$$\widetilde E(\widetilde U,\widetilde V)=\lim_{j\to\infty}\frac{E(\widetilde U_j,\widetilde V_j)}{j^n}.$$

This conjecture asserts that zero dipole moment is sufficient for the energy density of a periodically replicated configuration to converge to the infinite-periodic energy. The source presents it as a belief extending the asymptotics beyond compactly supported global minimizers, and provides no proof.