Maximality conjecture for the triangular and face-centred-cubic lattices in the grid cells model

Let $B_R$ be the radius-$R$ ball, let $\sigma_R$ be the uniform measure on $B_R$, and set $\alpha=\frac{10}{\pi}$. Define

$$R_2:=\frac{1}{2}\sqrt{\frac{2}{\sqrt{3}}},\qquad R_3:=2^{-\frac{5}{6}}.$$

Let $\mathcal{F}_{\sigma_R}^{\alpha}$ denote the grid-cells functional on unit-covolume lattices, and let $\mathsf{A}_2$ and $\mathsf{D}_3$ denote the triangular and face-centred-cubic lattices, respectively. Maximality conjecture. For all $R\leq R_2$, $\mathsf{A}_2$ is the unique maximizer of $\mathcal{F}_{\sigma_R}^{\alpha}$ in $\mathcal{L}_2(1)$, and for all $R\leq R_3$, $\mathsf{D}_3$ is the unique maximizer of $\mathcal{F}_{\sigma_R}^{\alpha}$ in $\mathcal{L}_3(1)$. These claims concern the grid cells problem in dimensions two and three and include the experimental regime $R=0.16$. The paper presents asymptotic and numerical evidence, while the rigorous study of $\mathcal{F}_{\mu}^{\alpha}$ remains challenging.