Conjecture on the three-dimensional vertex index

Let ${\bf K}$ be an arbitrary $0$-symmetric convex body in ${\mathbb R}^3$. A truncated octahedron of the form ${\bf T}-{\bf T}$ is obtained from an arbitrary tetrahedron ${\bf T}$ by Minkowski subtraction. Three-dimensional vertex-index conjecture.

$${\rm vein}({\bf K})\leq 12,$$

with equality for truncated octahedra of the form ${\bf T}-{\bf T}$, where ${\bf T}$ is an arbitrary tetrahedron in ${\mathbb R}^3$. The source gives only a conjectural upper bound beyond its established estimate $18$, so the assertion remains open.