Completeness conjecture for five-dimensional reflexive polytopes with $h^{1,1}=1$

A five-dimensional reflexive polytope is a lattice polytope whose only interior lattice point is the origin and whose polar dual is also a lattice polytope. Such a polytope gives rise to a four-dimensional Calabi--Yau hypersurface, whose Hodge number is denoted by $h^{1,1}$. Two polytopes are considered equivalent when they are non-isomorphic as lattice polytopes. Completeness conjecture. There are precisely $15$ inequivalent five-dimensional reflexive polytopes that give rise to four-dimensional Calabi--Yau hypersurfaces with Hodge number $h^{1,1}=1$. The conjecture is based on genetic-algorithm searches and comparison with the known partial classifications; completeness of the search is not established, so additional polytopes may exist.