The diameter bound and lattice-vector realization for primitive zonotopes

Let $\delta(d,k)$ denote the maximum diameter of a primitive lattice $(d,k)$-polytope. A Minkowski sum of lattice vectors is a zonotope obtained by summing line segments generated by lattice vectors.

Diameter conjecture.

$$\delta(d,k)\leq \left\lfloor\frac{(k+1)d}{2}\right\rfloor,$$

and $\delta(d,k)$ is achieved, up to translation, by a Minkowski sum of lattice vectors.

The preceding construction establishes this bound for the parameter range $k\leq 2d-1$, while the statement is posed generally for primitive zonotopes. The available context does not indicate whether the full claim is resolved.