The three-dimensional flatness conjecture for hollow convex bodies

Let a hollow convex $3$-body be a convex body in

$$whose interior contains no lattice points, and let$$

denote the flatness constant in dimension $3$, namely the supremum of the lattice widths of hollow convex $3$-bodies. Let

be the hollow $3$-simplex described above, of width $2+$. **Three-dimensional flatness conjecture.** No hollow convex $3$-body has width larger than

. That is,

$$\operatorname{Flt}(3)=2+\sqrt{2}.$$

This conjecture asserts that the displayed hollow simplex is extremal for lattice width among hollow convex bodies in dimension $3$. The supplied text does not indicate whether the claim has been proved or disproved.