Nill's facet-number conjecture for reflexive polytopes

Let ${\mathcal P}$ be a $d$-dimensional reflexive polytope, and let $N({\mathcal P})$ denote its number of facets.

Nill's conjecture.

$$N({\mathcal P}) \le 6^{d/2}.$$

This conjecture concerns a uniform upper bound on the number of facets of reflexive polytopes in terms of their dimension. The paper proves the bound for twinned chain polytopes, giving a partial answer; the general conjecture remains open.