McMullen's g-theorem for simplicial spheres

Let $\Sigma$ be a simplicial sphere of dimension $d-1$, with $f$-vector $(f_{-1},f_0,\ldots,f_{d-1})$ and associated $g$-vector defined by $g_k(\Sigma)=h_k-h_{k-1}$ for $k\leq d/2$, where $h_k=\sum_{i=0}^k(-1)^{k-i}\binom{d-i}{k-i}f_{i-1}$. An $M$-sequence is a sequence realized as the Hilbert function of a quotient of a polynomial ring by a homogeneous ideal. McMullen's g-theorem. A vector of $d$ integers is the $f$-vector of a simplicial sphere $\Sigma$ if and only if the associated $g$-vector is an $M$-sequence: there is a quotient $Q$ of a polynomial ring $\mathbb{R}[\mathbf{x}]$ by a homogeneous ideal such that

$$g_i(\Sigma)=\dim Q^i.$$

This characterizes the possible face vectors of simplicial spheres through the algebraic condition that their associated $g$-vectors are $M$-sequences. The statement is presented in the source as McMullen's conjecture; the supplied material does not establish its resolution.