Periodic Gaussian surface-area conjecture for periodic half spaces

Let $\Omega\subseteq\mathbb{R}^{n}$ be a periodic set and let $B\subseteq\mathbb{R}^{n}$ be a periodic half space. Periodic Gaussian surface-area conjecture. The Gaussian surface area of every periodic set is at least that of a periodic half space:

$$\int_{\partial\Omega}\gamma_{n}(x)\,dx\geq\int_{\partial B}\gamma_{n}(x)\,dx.$$

This is the $\rho\to1^{-}$ endpoint of the Khot–Moshkovitz periodic noise-stability conjecture. The paper derives it formally using the convergence of normalized noise stability to Gaussian surface area, but states that it studies rather than proves the endpoint conjecture.