Conjectures of Ene, Herzog, Hibi, and Saeedi Madani in the {sl Journal of Algebra}

In the preprint of ``Pseudo-Gorenstein and Level Hibi Rings,'' Ene, Herzog, Hibi, and Saeedi Madani assert (Theorem 4.3) that for a regular planar lattice $L$ with poset of join-irreducibles $P$, the following are equivalent:(1) $L$ is level;(2) for all $x,yin P$ such that $ylessdot x$, $height_{hat P}(x)+depth_{hat P}(y)lerank(hat P)+1$;(3) for all $x,yin P$ such that $ylessdot x$, either $depth(y)=depth(x)+1$ or $height(x)=height(y)+1$.They added, ``Computational evidence leads us to conjecture that the equivalent conditions given in Theorem 4.3 do hold for any planar lattice (without any regularity assumption).''Ene {sl et al.} prove the equivalence of (2) and (3) for a regular simple planar lattice, and write, ``One may wonder whether the regularity condition ... is really needed.''We show one cannot drop the regularity condition. Ene {sl et al.} say that ``we expect'' (2) to imply (1) for any finite distributive lattice $L$.We provide a counter-example.