3 problems

Neggers' conjecture. These $(P,\omega)$-Eulerian polynomials have only real roots.

Let $L$ be a distributive lattice, and let condition (b) in Theorem be the condition stated there for the levelness of the Hibi ring associated with $L$. Levelness conjecture. Cond…

Planar-lattice equivalence conjecture. The conditions (a), (b), and (c) given in Theorem hold equivalently for every planar lattice $L$.