Facet numbers of non-centrally symmetric reflexive polytopes arising from posets

Twinned chain polytopes form a broad class of non-centrally symmetric reflexive polytopes and exhibit intriguing structures. In the present paper, we show that the number of facets of $d$-dimensional twinned chain polytopes is at most $6^{d\mathrm{/}2}$. In case $d$ is even, the equality holds if and only if the polytope is isomorphic to a free sum of $d\mathrm{/}2$ copies of del Pezzo polygons. This result contributes a partial answer to Nill’s conjecture: the number of facets of a $d$-dimensional reflexive polytope is at most $6^{d\mathrm{/}2}$.