Conjecture on the maximal number of vertices of a reflexive polytope

Let $P$ be a $d$-dimensional reflexive polytope. Define ${\cal Z}_2:= \operatorname{conv}(\pm [0,1]^2)$, the centrally symmetric self-dual smooth Fano polytope with six vertices.

Vertex bound conjecture.

$$|{\cal V}(P)| \leq 6^{\frac{d}{2}},$$

where equality occurs if and only if $d$ is even and $P \cong ({\cal Z}_2)^{\frac{d}{2}}$.

The conjecture seeks a sharp dimension-dependent bound on the number of vertices of a reflexive polytope, equivalently on the rank of the class group of the associated toric variety. The stated parser status is unknown, so its resolution is not established by the supplied source.