The $g$-conjecture for simplicial and rational homology spheres

Let $\Delta$ be a simplicial sphere or a rational homology sphere. Its $h$-vector is subject to the $g$-theorem conditions: Dehn–Sommerville symmetry, unimodality up to the middle, and the pseudopower inequalities making its $g$-vector an $M$-vector. The $g$-conjecture. The $g$-theorem holds for all simplicial spheres, or even for all rational homology spheres. This is the main open problem in the theory of face enumeration; the paper proves it for PL homology spheres, but the assertion for arbitrary simplicial or rational homology spheres remains open.