The second and fourth authors have conjectured that a certain hollow tetrahedron $Δ$ 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 $Δ$ in the Hausdorff distance such that every convex body in $U \setminus \{Δ\}$ has width strictly smaller than $Δ$. 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 $[2.653, 19.919]$.