McMullen's $g$-conjecture for homology spheres

A homology sphere is a pure simplicial complex $L$ such that every face link has the homology of a sphere of the same dimension. Its $g$-vector is the sequence derived from its $h$-vector in the usual way, and an $M$-sequence is the $f$-vector of a multicomplex. McMullen's $g$-conjecture. The $g$-vector of every homology sphere is an $M$-sequence. This is the well-known $g$-conjecture for spheres, originally raised by McMullen for simplicial spheres; the source gives no resolution evidence, so its status is left open.