The orbit lattice conjecture for platycosms

A platycosm $P$ is a compact flat three-dimensional manifold. Its orbit lattice bound $R_0$ is the maximum of the covering radii of all orbit lattices for $P$, and $P$ has the orbit lattice property if its diameter equals $R_0$.

Orbit lattice conjecture for platycosms. All 10 platycosms have the orbit lattice property.

The orbit lattice bound gives a lower bound for the diameter of every platycosm; when an orbit lattice has a translatable Voronoi tiling, the bound is attained. The conjecture asserts that this equality holds for all 10 platycosms.