Conjecture on the maximal number of vertices of a simplicial reflexive polytope

Let $P$ be a $d$-dimensional simplicial reflexive polytope, meaning that the associated toric variety is ${\mathbb Q}$-factorial. Let ${\cal V}(P)$ denote its set of vertices, let $P^*$ be its dual polytope, and let ${\cal Z}_2:= \operatorname{conv}(\pm [0,1]^2)$. Also let $S_3$ be the nonsingular toric del Pezzo surface associated with ${\cal Z}_2$.

Simplicial vertex bound conjecture.

$$|{\cal V}(P)| \leq \begin{cases} 3d, & d \text{ even},\\ 3d-1, & d \text{ odd}. \end{cases}$$

For $d$ even, equality holds if and only if $P^* \cong ({\cal Z}_2)^{\frac{d}{2}}$, equivalently $X \cong (S_3)^{\frac{d}{2}}$.

This is a proposed sharp bound for simplicial reflexive polytopes, whose associated toric varieties are ${\mathbb Q}$-factorial and whose class number equals the Picard number. The stated parser status is unknown, so the conjecture remains open in this record.