Bj1rner and Swartz's $g$-vector conjecture for doubly CohenMacaulay complexes

Let $X$ be a doubly CohenMacaulay, or $2$-CM, complex, and let $g(X)$ denote its $g$-vector. An $M$-sequence is the $f$-vector of an order ideal of monomials. Bjrner and Swartz's $g$-vector conjecture. The $g$-vector of any $2$-CM complex is an $M$-sequence. The paper proves that the initial part $(g_0,g_1,g_2)$ is an $M$-sequence for $2$-CM complexes of dimension at least $3$, but the assertion for the entire $g$-vector is presented as a conjecture.