Lattice-width conjecture for lattice k-point tetrahedra

Let $T$ be a lattice tetrahedron with exactly $k$ interior lattice points. In a minimal direction for its lattice width, consider the consecutive lattice planes meeting $T$. Lattice-width conjecture. The lattice width of $T$ is at most

$$k+1,$$

and there is at least one interior lattice point on each of the consecutive lattice planes in any minimal direction. The preceding theorem gives a general upper bound of order $n^{1/3}$, and the examples in the source show that width need not increase with the number of interior points; the conjectured bound and plane-by-plane interior-point condition remain open.