A counterexample to the facet bound for reflexive polytopes

A lattice polytope is reflexive if it contains the origin in its interior and its polar is also a lattice polytope. We construct a $45$-dimensional reflexive lattice polytope with $2^{32}\binom{40}{8}+5$ facets, exceeding $6^{45/2}$. This disproves the unrestricted facet bound conjectured by Nill. The construction starts with a lattice polytope having many vertices and a polar whose denominators divide $32$. Five successive pyramids multiply the original polar coordinates by $32$ while adding only five vertices. Taking the polar gives the required counterexample.