Strong dimension conjecture for dual cells

Let $D$ be a dual $k$-cell, and let a Delone tessellation of a lattice be its tessellation into Delone cells.

Strong dimension conjecture. For every dual $k$-cell $D$ there is a $k$-dimensional lattice $\Lambda$ such that there is a $k$-dimensional cell in the Delone tessellation of $\Lambda$ equivalent to $D$.

This is presented as a stronger version of the dimension conjecture. Both conjectures are stated as open, and a counterexample would immediately give a counterexample to the Voronoi conjecture.