The Björner–Swartz conjecture for g-vectors of doubly Cohen–Macaulay complexes

Let $K$ be a simplicial complex that is doubly Cohen–Macaulay over a fixed field $k$, meaning that $K$ is Cohen–Macaulay and, for every vertex $v otin K$, the deletion $K\setminus v$ is Cohen–Macaulay of the same dimension as $K$. The $g$-vector of $K$ is the sequence associated with its face numbers.

Björner–Swartz conjecture. The $g$-vector of any doubly Cohen–Macaulay complex is an $M$-sequence.

The conjecture extends the known $M$-sequence results for homology spheres and other classes of doubly Cohen–Macaulay complexes. The source attributes it to Björner and Swartz and notes that it weakens an earlier problem.