Gorenstein* poset nonnegativity and Boolean-lattice lower-bound conjectures

A poset is Gorenstein* when it is Cohen--Macaulay and Eulerian. Its $cd$-index is denoted by $\Phi_P(c,d)$; coefficientwise comparison means comparison of the coefficients of corresponding $cd$-words. Gorenstein poset conjectures.*

• The $cd$-index of every Gorenstein* poset is nonnegative.
• The $cd$-index of every Gorenstein* lattice is coefficientwise greater than or equal to the $cd$-index of the Boolean lattice (simplex).

These conjectures seek positivity for the broad class of Gorenstein* posets and a stronger lower bound for Gorenstein* lattices. The source records earlier nonnegativity results for shellable CW spheres and gives no resolution of either assertion.