Hexagonal-lattice conjecture for simultaneous quantum optimization

Let $\Lambda$ range over periodic configurations of arbitrary fixed density in phase space, and consider the associated quantum packing, covering, and paving problems. The hexagonal-lattice conjecture. The hexagonal lattice solves all quantum problems simultaneously among periodic configurations of arbitrary fixed density. The conjecture is presented as a heuristic generalization of the known two-dimensional hexagonal-lattice extremality results, but no proof is given for the full collection of quantum problems and arbitrary fixed density.