Equal-density conjecture for p2mg, cm, and p4 packings

Let $n$-gons be regular polygons, and write the density of the densest packing with plane group symmetry $G$ as the corresponding densest $G$-packing density. An $n$-gon has a $k$-fold rotational symmetry if rotation by an angle of $2\pi/k$ preserves it, and $k\in\mathbb{N}$ means that $k$ is a positive integer. p2mg–cm–p4 equal-density conjecture. The densities of the densest $p2mg$ and $cm$ packings are equal for all $n$-gons except $n$-gons with $(12k-1)$-fold and $(12k+1)$-fold rotational symmetry, where $k\in\mathbb{N}$; and the densities of the densest $p2mg$, $cm$, and $p4$ packings are equal for all $n$-gons containing a twelve-fold rotational symmetry. These experimentally motivated equalities remain unresolved in the source.