  The second and fourth authors have conjectured that a certain hollow
tetrahedron $\Delta$ of width $2+\sqrt2$ attains the maximum lattice width
among all three-dimensional convex bodies. We here prove a local version of
this conjecture: there is a neighborhood $U$ of $\Delta$ in the Hausdorff
distance such that every convex body in $U \setminus \{\Delta\}$ has width
strictly smaller than $\Delta$. When the search space is restricted to
tetrahedra, we compute an explicit such neighborhood. We also limit the space
of possible counterexamples to the conjecture. We show, for example, that their
width must be smaller than $3.972$ and their volume must lie in $[2.653,
19.919]$.
