Parallelotope covering beyond the established cases

Determine whether every parallelotope in n that contains more than half of the vertices of the unit cube [0,1]^n satisfies P+n=n.

Background

The paper studies when a parallelotope containing vertices of the unit cube is a lattice covering under integer translations. It notes that the threshold of more than half the cube vertices cannot be lowered, and establishes the affirmative answer in dimension 2 and for parallelepipeds containing five vertices of the three-dimensional cube. Theorem 2 provides a partial affirmative result in arbitrary dimension when the parallelotope contains an entire facet of the cube together with the opposite vertex, but the general question remains unresolved in the paper.

References

Does every parallelotope $P\subset\Bbb Rn$ containing more than half of the vertices of $Q_n$ satisfy $P+\Bbb Zn=\Bbb Rn$?

— Lattice coverings by parallelotopes containing cube vertices  (2610.21572 - Grozev et al., 15 Oct 2026) in Section 1, Results, immediately before Proposition 1