The $g$-conjecture for rational homology spheres

Let a rational homology sphere be a simplicial complex whose homology with rational coefficients is that of a sphere, and let its $g$-vector be the vector obtained from its $h$-vector by the usual transform

$$g_i=h_i-h_{i-1}$$

for the relevant indices. An $M$-vector is an integer sequence satisfying Macaulay's numerical conditions.

The $g$-conjecture. The $g$-vector of a rational homology sphere is an $M$-vector.

This conjecture generalizes the characterization of face numbers of simplicial polytopes. It is known for polytopal spheres, but the source presents the assertion for arbitrary rational homology spheres as the motivating conjecture.