Dimension conjecture for dual cells

Let $D$ be a dual $k$-cell, and let its affine span be the smallest affine subspace containing $D$.

Dimension conjecture. For every dual $k$-cell, the dimension of its affine span is equal to $k$.

The conjecture is known for $k\leq 3$ because all dual $3$-cells are classified. A counterexample would immediately give a counterexample to the Voronoi conjecture, while the assertion remains open in higher dimensions.