Vertex bound conjecture for primitive polytopes

Let $P$ be a primitive polytope in $\mathbb{E}^d$, where $d\geq 4$. Primitive-polytope vertex conjecture. The polytope $P$ has at most $2^d$ vertices, and it has fewer than $2^d$ vertices if it is not an affine $d$-cube. This conjecture is posed as a main consequence of the chapter's study of primitive polytopes and would provide a sharp cube-type vertex bound in dimensions at least four. The supplied excerpt gives no further evidence about its resolution.