Conjecture on equal densities of densest plane group packings

Let an $n$-gon be a regular polygon, and let the density of its densest packing with plane-group symmetry $G$ be denoted by the corresponding packing density. The relevant classes are centrally nonsymmetric $n$-gons with three-fold rotational symmetry, centrally symmetric $n$-gons, and $n$-gons containing a six-fold rotational symmetry.

Equal-density packing conjecture. Densities of the densest $p2$, $pg$, and $p2gg$ packings are equal for all, but centrally nonsymmetric $n$-gons with three-fold rotational symmetry and $n\geq 9$, densities of the denses $p2$, $pg$, $p2gg$, and $p1$ packings are equal for all centrally symmetric $n$-gons, and densities of the densest $p2$, $pg$, $p2gg$, $p1$, and $p3$ packings are equal for all $n$-gons containing a six-fold rotational symmetry.

These conjectured equalities summarize the observed exceptional behavior of three-fold and six-fold rotational symmetries in densest plane-group packings of regular polygons. The supplied text does not state which cases have been proved or remain open.