We examine a concrete smooth Fano 5-polytope $P$ with 8 vertices with the following properties: There does not exist a smooth Fano 5-polytope $Q$ with 7 vertices such that $P$ contains $Q$, and there does not exist a smooth Fano 5-polytope $R$ with 9 vertices such that $R$ contains $P$. As the polytope $P$ is not pseudo-symmetric, it is a counter example to a conjecture proposed by Sato.
"Smooth Fano polytopes can not be inductively constructed." Tohoku Math. J. (2) 60 (2) 219 - 225, 2008. https://doi.org/10.2748/tmj/1215442872